Università di Tor Vergata – Dipartimento di Ingegneria Civile ed Ingegneria Informatica
Exercises in Mathematical Analysis I
Alberto Berretti, Fabio Ciolli
2
1 Fundamentals
1.1 Polynomial inequalities
Solve the following inequalities for x ∈ R:
h
Ex. 1. (x3 − 3x + 2)(x − 4) > 0.
h
Ex. 2. (1 − x)(x − 3)(x + 2) < 0.
x < −2, x > 4
i
− 2 < x < 1, x > 3
i
1.2 Rational inequalities
Solve the following inequalities for x ∈ R:
Ex. 3.
x+1
x−1
x2 + x − 2
+3
.
<
x−7
x2 − 10x + 21 x − 3
Ex. 4.
x + 12
x−6
3x − 3
− 2
≥
.
x+8
x−6
x + 2x − 48
Ex. 5.
−9x2 − 12x − 4
< 0.
2x2 − 5x + 2
Ex. 6.
(x − a)(x − b)
≥ 0, a > b > 0.
x2 − a2
h
i
x < 0, 3 < x < 5, x > 7
h
−8<x<6
i
i
h
1
2 2
x<− ,− <x< ,x>2
3 3
2
h
i
x < −a, b ≤ x < a, x > a
1.3 Irrational inequalities
Solve the following inequalities for x ∈ R:
Ex. 7. 2x − 3 >
Ex. 8. x − 8 <
Ex. 9.
√
√
h
i
x≥3
4x2 − 13x + 3.
h
i
x ≤ 2, x ≥ 7
x2 − 9x + 14.
√
√
x − 1 − x − 2 < 2.
√
Ex. 10.
√
Ex. 11.
x+2<8+
3x − 8 >
√
h
i
x≥2
√
h
i
x≥6
x − 6.
5x + 3 +
√
h
i
no solution
x + 6.
√
h
5 + 5i
x≥
2
√
Ex. 12.
x − 1 ≤ x − 2.
3
A. Berretti, F. Ciolli
Exercises in Mathematical Analysis I
√
h
i
x≥1
x − 1 ≥ −100 − x.
Ex. 13.
√
x−2
< 1.
Ex. 14. √
x−4
√
3
Ex. 15. |x + 8| > 1.
√
Ex. 16.
4 − |x + 3| < 2.
Ex. 17.
√
3
4 − |x + 3| < 2.
√
Ex. 18.
√
Ex. 19.
h
i
2 ≤ x < 16
h
i
x < −9, x > −7
h
−7≤x≤1
i
h i
R
h −5 +
4 − |x + 2| < 2 − |x|.
2
√
17
<x<1
i
h
1i
0<x≤
4
3 − |4x + 2| < 1 − 2|x|.
1.4 Absolute value inequalities
Solve the following inequalities for x ∈ R:
h
i
{x ≤ −2} ∪ {x ≥ 4}
Ex. 20. | |x − 1| − 1 | ≥ 2.
Ex. 21. |x − 2| − |x| < 3.
h i
R
Ex. 22. | |x − 2| − |x| | ≤ 3.
h i
R
Ex. 23.
|x2
− 2x − 4| ≥ |x| + 2.
Ex. 24. |x − 2| + |x| < 3.
x − 2 Ex. 25. − |x − 2| < 2.
x − 3
h
√
√ ) (
(
)
i
3 + 33
3 − 17
x ≤ −2} ∪ {x ≥
∪
≤x≤2
2
2
h 1
5 i
− <x<
2
2
h
√ i
√
{x < 1 + 3} ∪ {x > 2 + 2}
1.5 Exponential and logarithmic inequalities
Solve the following inequalities for x ∈ R:
h
2 log 3 i
x<
log 108
Ex. 26. 4x+1 63x−2 < 8x .
h
log 3
7
7i
x< −
,x>
2 2 log 5
2
h i
R
Ex. 27. 3 · 52(2x−7) − 4 · 5(2x−7) + 1 > 0.
Ex. 28. log3 (2x2 − 7x + 103) > 2.
4
A. Berretti, F. Ciolli
Ex. 29. log5
(x2
Exercises in Mathematical Analysis I
√
√
h
i
7− 5 7+ 5
2<x<
,
<x<5
2
2
h 10
i
−
< x < 2, x > 3
13
h
i
0<x<2
− 7x + 11) < 0.
Ex. 30. log10 (x + 4)2 > log10 (13x + 10).
Ex. 31. 22x − 5 · 2x + 4 < 0.
Ex. 32.
2x
3
2
6
+ x
> x
+ 5.
−1 2 +1 2 −1
h
i
0<x<1
h
Ex. 33. | log10 (3x + 4) − log10 7| < 1.
−
i
11
< x < 22
10
1.6 Trigonometric inequalities
Solve the following inequalities for x ∈ R:
Ex. 35.
Ex. 36.
Ex. 37.
Ex. 38.
hπ
i
5
+ 2kπ < x < π + 2kπ, π + 2kπ < x < π + 2kπ, k ∈ Z
3
3
hπ
i
5
cos 2x + 3 sin x ≥ 2.
+ 2kπ ≤ x ≤ π + 2kπ, k ∈ Z
6
6
i
h π
√
π
π
π
3 tan2 x − 4 3 tan x + 3 > 0.
− + kπ < x < + kπ, + kπ < x < + kπ, k ∈ Z
2
6
3
2
1
loga
− | sin x| < 0, a > 1.
2
h 1
i
1
5
7
− π + 2kπ < x < π + 2kπ, π + 2kπ < x < π + 2kπ, k ∈ Z
6
6
6
6
h
i
3 cos x + sin2 x − 3 > 0.
not possible
Ex. 34. 2 sin2 x − cos x − 1 > 0.
h 7
i
√
π
π
− π + 2kπ ≤ x ≤ + 2kπ, k ∈ Z
Ex. 39. 4 cos x +
− 2 3 cos x + 1 ≥ 0.
6
6
6
hπ
i
5
7
11
cos 2x Ex. 40. ≤
1.
+
2kπ
≤
x
≤
π
+
2kπ,
π
+
2kπ
≤
x
≤
π
+
2kπ,
k
∈
Z
sin x 6
6
6
6
h
i
π
5
tan 2x < 1.
kπ < x < + kπ, π + kπ < x < π + kπ, k ∈ Z
Ex. 41. cot x
6
6
1.7 Boundedness of numerical sets
Study the boundedness of the following numerical sets, expressing for any of them sup, inf, max
and min by verifying the definition
1
Ex. 42. A = 2
,n∈N .
n +1
(
)
(−1)n
,n∈N .
Ex. 43. A =
n2 + 2
h
h
5
inf A = 0, max A = 1
i
1
1i
min A = − , max A =
3
2
A. Berretti, F. Ciolli
x+2
, x ∈ R, x > 3 .
x−3
x+2
A=
, x ∈ R, x < 2 .
x−2
nm
A= 2
, (n, m) ∈ N × N \ {(0, 0)} .
n + m2
nm
A= 2
, (n, m) ∈ N \ {0} .
n + m2
n+m
A=
, n, m ∈ N, n , m .
n−m
n m
+ , n, m ∈ N \ {0} .
A=
m n
Ex. 44. A =
Ex. 45.
Ex. 46.
Ex. 47.
Ex. 48.
Ex. 49.
Exercises in Mathematical Analysis I
h
inf A = 1, sup A = +∞
i
h
inf A = −∞, sup A = 1
i
h
min A = 0, max A =
h
h
inf A = 0, max A =
1i
2
1i
2
inf A = −∞, sup A = +∞
h
inf A = 2, sup A = +∞
i
i
Study the boundedness of the following numerical sets, expressing for any of them
sup, inf, max and min
3n + 1
, n ∈ N \ {0} .
Ex. 50. A =
n+2
1
Ex. 51. A =
, n ∈ N \ {0} .
1 + 2−n
2n
Ex. 52. A =
, n ∈ N \ {0} .
n! + 1
(
)
log n!
Ex. 53. A =
,n∈N .
n!
(
)
n
Ex. 54. A =
,n∈N .
sin(1 + nπ/2)
√

√



 n− n+2
,
n
∈
N
\
{0}
Ex. 55. A = 


.
n2
n
1 Ex. 56. A = (−1)n
− , n ∈ N .
n+3 5
nπ 2
Ex. 57. A = n + sin( ) , n ∈ N .
2
(
!
)
(2n + 1)π 1/(n+1)
Ex. 58. A = sin
2
,n∈N .
2
i
4
min A = , sup A = 3
3
h
i
2
min A = , sup A = 1
3
h
4i
inf A = 0, max A =
3
h
√ i
min A = 0, max A = log 2
h
h
h
inf A = −∞, sup A = +∞
min A = −
2
√ , sup A = 0
1+ 3
i
i
1
6i
min A = , sup A =
5
5
h
i
min A = 0, sup A = +∞
h
h
i
√
min A = − 2, max A = 2
Establish if the following numerical sets are bounded; find sup, inf, max and min, if they
exist
1
, n ∈ N, n ≥ 1 .
1 + 2n
x
1
Ex. 60. A = x ∈ R :
>
.
x+1 2
Ex. 59. A =
1i
3
h
i
A = (−∞, −1) ∪ (1, +∞); inf A = −∞, sup A = ∞
h
6
inf A = 0, max A =
A. Berretti, F. Ciolli
√
1
Ex. 61. A = x ∈ R : x2 − 2x < x .
2
p
Ex. 62. A = {x ∈ R : log(sin x) ∈ R}.
Ex. 63. A = {x ∈ R : 1 ≤ 32x+1 < 9}.
(
)
1 3x−3
Ex. 64. A = x ∈ R : 5 <
≤ 25 .
5
(
)
(−1)n
Ex. 65. A = 1 −
, n ∈ N \ {0} .
n

4


,
n ∈ N, n even



2n
+1


Ex. 66. A = 





 2 − 1 , n ∈ N, n odd
n+1
Exercises in Mathematical Analysis I
8i
3
h
i
π
A = { + 2kπ, k ∈ Z}; inf A = −∞, sup A = +∞
2
h
1
1i
min A = − , sup A =
2
2
h
1
2i
min A = , sup A =
3
3
h
h








.







min A = 2, sup A =
i
1
min A = , max A = 2
2
h
inf A = 0, max A = 4
i
Ex. 67. Define an infinite set using a non-monotone sequence such that 0 and 1 will be the
inf and sup of the set respectively.
Ex. 68. Find inf e sup of the areas of the surfaces of the rectangles with perimeter equal to
4a, for a a positive real number, different from zero.
1.8 Domain of functions
Determine the domain of the following functions and study the boundedness of such sets.
Then trace a qualitative graph of the functions themselves.
√
Ex. 69. f (x) = x2 − 1.
r
1−x
Ex. 70. f (x) =
.
x+2
r
4 |1 − x|
Ex. 71. f (x) =
.
x+2
Ex. 72. f (x) = log1/2 (1 − |x|).
q
Ex. 73. f (x) = 6 log1/3 (2 − |x|).
Ex. 74. f (x) =
q
Ex. 75. f (x) =
p
log2 (x2 − 2x − 5) − 1.
log3 (2x + 2) − log3 x.
r
x+2
Ex. 76. f (x) = log3 (
).
x
7
A. Berretti, F. Ciolli
Ex. 77. f (x) =
p
Exercises in Mathematical Analysis I
log3 (x + 1) − log9 (x + 2) + 1.
2 −3x−4)
Ex. 78. f (x) = 2(x+2)/(x
.
Ex. 79. f (x) = log5 (62x − |4 · 6x − 1|).
2x − 1
.
Ex. 80. f (x) = cos
x+1
r
2x − 1
Ex. 81. f (x) = cos
.
x+1
2x − 1
1 1/4
.
Ex. 82. f (x) = cos
−
x+1
2
Ex. 83. f (x) =
1
.
sin x + cos x
Ex. 84. f (x) = 2 log3 (sin x + 2 cos x).
Ex. 85. f (x) = log3 (sin x + 2 cos x)2 .
Ex. 86. f (x) = log23 (sin x + 2 cos x).
x+1
Ex. 87. f (x) = arccos
.
x−1
x+1
Ex. 88. f (x) = arcsin
.
|x| − 1
1/2
Ex. 89. f (x) = log4 (sin x)
.
1/2
√
4
2
.
Ex. 90. f (x) = 2 1−log7 (x +x) − (x2 + x)
Ex. 91. Indicated by D the domain of any function of the exercises in the paragraph 1.8,
determine the set of the interior points D̊ of D and the set of its boundary points ∂D.
Moreover, say if such sets are oper or closed and study their boundedness.
Ex. 92. Determine the set of the images (range) for any function of the exercises in the
paragraph 1.8, and the set of the accumulation points of such sets.
Ex. 93. Given two functions f, g : A ⊆ R → R, show the following implications:
1. f, g increasing =⇒ f + g increasing;
2. f, g decreasing =⇒ f + g decreasing;
3. f increasing and g strictly increasing =⇒ f + g strictly increasing;
4. f decreasing and g strictly decreasing =⇒ f + g strictly decreasing.
8
A. Berretti, F. Ciolli
Exercises in Mathematical Analysis I
Ex. 94. Establish under which conditions the following implication is true:
f, g increasing (or decreasing) =⇒ f · g increasing (or decreasing).
Ex. 95. Furnish an example such that the result of the exercise 94 is, in general i.e. without
further hypothesis, false.
Ex. 96. Show that if f : A ⊆ R → R is invertible, then
f increasing (decreasing) =⇒ f −1 increasing (decreasing).
Ex. 97. Let f : A ⊆ R → R be such that 0 < f (A) and increasing. Determine if
1
is increasing
f
or decreasing.
Ex. 98. Let f, g : A ⊆ R → R two injective functions. Is the function f + g invertible?
Ex. 99. Let f : X → Y and g : V → W and let moreover f (X) ∩ V , ∅. If f and g are invertible
functions, is the composition f ◦ g an invertible function?
Ex. 100. Furnish three different examples of functions f : X → X such that f ≡ f −1 .
1.9 Invertibility of functions
Study the invertibility of the following functions in their natural definition set.
Ex. 101. f (x) = 2x + x.
Ex. 102. f (x) = −x + log1/2 x.
Ex. 103. f (x) = x2 + log3 (1 + x).
Ex. 104. f (x) =
5x
+ x3 .
1 + 5x
Ex. 105. f (x) = x|x| + 1.

1



if x > 1

x
−
1
Ex. 106. f (x) = 
al variare di a ∈ R.


 x + a if x ≤ 1


2


 x + ax if x ≤ 0
Ex. 107. f (x) = 
for any a ∈ R.
1


if x > 0
 −
x


3

 x if |x| ≥ 1
Ex. 108. f (x) = 
for any a ∈ R.

 ax if |x| < 1.
9
A. Berretti, F. Ciolli
Exercises in Mathematical Analysis I
Ex. 109. Let f : X → Y and g : V → W be two invertible functions such that it is well defined
the composed function g ◦ f . Call f −1 and g−1 their inverses respectively, show that
(g ◦ f )−1 = f −1 ◦ g−1 .
Verify that the following functions are invertible; then determine the inverse of any of
them, specifying its domain.
Ex. 110. f (x) = x|x| + x.
Ex. 111. f (x) = x(x − 2), x ≤ 0.
Ex. 112. f (x) = log1/2 (1 − x3 ).
3x+1
.
1 + 3x+1
√
Ex. 114. f (x) = e2x + ex + 1.
!
x2
3
Ex. 115. f (x) = sin
, x ≤ 0.
x2 + 1
Ex. 113. f (x) =
Ex. 116. f (x) = arccos(log2 x).
Ex. 117. f (x) = tan(x3 + 1),
3
π
< x3 + 1 < π.
2
2
Ex. 118. f (x) = arctan(x3 + 1).
√
Ex. 119. f (x) = arcsin( x2 + 1),
x < 0.
10
2 Complex numbers
2.1 Elementary properties of complex numbers
Determine z̄, Im(z) e |z| in the following cases:
h
i
3 + 4i, −4, 5
Ex. 120. z = 3 − 4i.
h
Ex. 121. z = (2 − i)(−3 + 2i).
h
Ex. 122. z = (1 − i)(3 − 7i).
1 − 2i
2
Ex. 124. z =
Ex. 125. z =
2
.
3−i
Ex. 126. z =
2 − 3i
.
1−i
2
i
− 5 + 12i, −12, 13
3
5i
+ i, −1,
4
4
r
h3 − i 1
2i
, ,
5 5
5
r
h5 + i 1
13 i
,− ,
2
2
2
h
√ i
− 11 − 2i, 2, 5 5
h
.
Ex. 127. z = (1 − 2i)3 .
−
h −2 + 6i
(1 − i)3
Ex. 128. z =
.
2−i
Ex. 129. z =
i
65
√ i
− 4 + 10i, −10, 2 29
h
Ex. 123. z = (2 − 3i)2 .
√
− 4 − 7i, 7,
5
(1 + 2i)4
.
(1 − i)2
6
,− ,2
5
r
2i
5
h
7
7 25 i
12 + i, − ,
2
2 2
Compute the absolute value and argument of z in the following cases:
h√ π
Ex. 130. z = 1 + i.
2,
4
h
√
π
Ex. 131. z = 1 − 3i.
2, −
3
√
h2 π
3 i
Ex. 132. z =
+ .
,
3
3
3 6
h√ π
Ex. 133. z = i(1 − i).
2,
4
h
2
π
Ex. 134. z =
1, −
√ .
3
1 + 3i
11
+ 2kπ, k ∈ Z
i
+ 2kπ, k ∈ Z
i
+ 2kπ, k ∈ Z
i
+ 2kπ, k ∈ Z
i
+ 2kπ, k ∈ Z
i
A. Berretti, F. Ciolli
Ex. 135. z = √
3
3+i
Exercises in Mathematical Analysis I
i
π
+ 2kπ, k ∈ Z
2
6
h √
i
π
2 2, − + 2kπ, k ∈ Z
12
h 1
i
5
√ , − π + 2kπ, k ∈ Z
2 12
h3
.
√
Ex. 136. z = (1 − i)( 3 + i).
1−i
.
Ex. 137. z = √
3+i
√
3−i
Ex. 138. z =
√ .
1 + 3i
,−
i
h
π
1, − + 2kπ, k ∈ Z
2
h
i
64, π + 2kπ, k ∈ Z
Ex. 139. z = (1 − i)12 .
!14
.
h 214 π
i
,
+
2kπ,
k
∈
Z
37 3
Ex. 141. z =
i324 − i261
.
i145 + i492
Ex. 142. z =
1 − i1039
.
i2048 − i1457
Ex. 143. z =
cos 2θ − i sin 2θ
.
sin θ + i cos θ
h
i
π
1, − + 2kπ, k ∈ Z
2
h π
i
1, + 2kπ, k ∈ Z
2
h
i
π
1, −θ − + 2kπ, k ∈ Z
2
i
Ex. 140. z = 1 + √
Ex. 144. z =
1
2
3
sin 2θ + i cos θ
.
1 − i sin
i
h θ
π
π
| cos θ|, se cos θ > 0, − se cos θ < 0, not determinate if cos θ = 0
2
2
2.2 Roots of complex numbers
Compute the following roots of the complex numbers:
q
√
Ex. 145. 1 + 3i.
q
√
3
Ex. 146. 1 + 3i.
r
Ex. 147.
Ex. 148.
p
4
Ex. 149.
3 + ii
√
2
h√
π
π √
7π
7π
3
3
2 cos + i sin
, 2 cos
+ i sin
,
9
9
9
9
√
5π
5π i
3
2 cos
− i sin
9
9
1+i
.
1−i
±
1 + ii
± √
2
h
i
1 + i, 1 − i, −1 + i, −1 − i
h
(1 − i)3 + (1 + i)3 .
s
√
h
q
i)2
(1 −
2
.
(1 + i)3
h √
3π
3π
4
± 2 cos
+ i sin
=±
8
8
12
√
2−1+i
√
2
q
√
2 + 1i
A. Berretti, F. Ciolli
Exercises in Mathematical Analysis I
Compute the following expressions containing complex numbers:
s
h 1 + ii
(1 + i)2 − (1 − i)3 − 2
Ex. 150.
.
±
2
(3 − i)2 + 6i
r
√
h
3
(2 − 3i)2 + 7i
3 ii
1+
Ex. 151.
.
i, ±
−
5
2
2
r
h 1 + i 1 − i −1 + i −1 − i i
(1 − i)(1 + 3i)
4 (1 + 2i)(3 − i)
Ex. 152.
−
.
√ , √ , √ , √
5
2
2
2
2
2
s
√
h√
(2 − i)2 − 3
−1 ± 3 i
3
3
Ex. 153.
.
2,
√
3
3
(1 + 2i) + 11
4
√
√
h
√
√
3i
3
i
3i
3
0, −2i,
±
,− ±
Ex. 154. −1 + i.
2
2
2
2
2.3 Complex equations
2.3.1 Algebraic complex equations
Determine the solutions of the following algebraic equations:
h
i
z = −1 + i, −1 − i
h
i
z = 3 + 2i, 3 − 2i
Ex. 155. z2 + 2z + 2 = 0.
Ex. 156. z2 − 6z + 13 = 0.
h
i
1
1
z = + 2i, − 2i
2
2
h
i
z = 1, −2 + i, −2 − i
h
i
z = 1 + i, 1 − i, −1 + i, −1 − i
h
i
z = −2i, 3i
Ex. 157. 4z2 − 4z + 17 = 0.
Ex. 158. z3 + 3z2 + z − 5 = 0.
Ex. 159. z4 + 4 = 0.
Ex. 160. z2 − iz + 6 = 0.
1
ii
z= ,−
2 2
h
3i i
z = i, −
2
h
ii
z = 2i,
2
h
1 ii
z= ,
2 3
h
i
ii
z=1+ , 3−
2
4
h
i
i
z = 1, , −i
2
h
ii
z = −1, −3i,
2
h
Ex. 161. 4z2 − 2(1 − i)z − i = 0.
Ex. 162. 2z2 + iz + 3 = 0.
Ex. 163. 2z2 − 5iz − 2 = 0.
Ex. 164. 6z2 − (3 + 2i)z + i = 0.
Ex. 165. 8z2 − 2(16 + i)z + 5(5 + 2i) = 0.
Ex. 166. 2z3 − (2 − i)z2 + (1 − i)z − 1 = 0.
Ex. 167. 2z3 + (2 + 5i)z2 + (3 + 5i)z + 3 = 0.
13
A. Berretti, F. Ciolli
Exercises in Mathematical Analysis I
2.3.2 Non-algebraic complex equations
Ex. 168. z|z3 | + 16 = 0.
Ex. 169. z2 |z2 | + 16 = 0.
Ex. 170. z3 |z| + 16 = 0.
Ex. 171. z2 (1 + |z2 |) = −20.
Ex. 172. z2 (1 − |z2 |) = −20.
Ex. 173. z2 (4 − |z2 |) = 5.
Ex. 174. z2 (4 − |z2 |) = 4.
Ex. 175. z2 (4 − |z2 |) = 3.
h
i
z = −2
h
i
z = 2i, z = −2i
h
√ i
√
z = −2, z = 1 + 3i, z = 1 − 3i
h
i
z = 2i, z = −2i
h
√
√ i
z = 5, z = − 5
h
√
√ i
z = 5i, z = − 5i
q
q
h
√
√
√ i
√
z = 2i, z = − 2i 2 + 2 2i, − 2 + 2 2i
q
q
h
√
√ i
√
√
z = 1, z = −1, z = 3, z = − 3, z = 2 + 7i, − 2 + 7i
Ex. 176.
z2
1
=− .
2
2
1 + |z |
h
i
z = i, z = −i
Ex. 177.
z2
= −2.
1 + |z2 |
h
i
Nessuna soluzione
z2
1
Ex. 178.
=− .
2
1 − |z2 |
Ex. 179.
z2
= −2.
1 − |z2 |
Ex. 180.
z4
== −8.
|z2 |
Ex. 181.
1
z2
== − .
4
8
|z |
Ex. 182.
z4
== 8.
|z6 |
Ex. 183.
z2 − |z2 |
+ 1 = 0.
4 + |z2 |
Ex. 184.
z2 − |z2 | 1
+ = 0.
2
4 + |z2 |
Ex. 185.
z2 − |z2 | 1
+ = 0.
4
4 + |z2 |
r
h
h
z=
2
i, z = −
3
r
2i
i
3
h
i
i i
z= √ ,z=−√
3
3
h
i
z = 2 + 2i, z = 2 − 2i, z = −2 + 2i, z = −2 − 2i
h
z=
√
√ i
z = 2 2i, z = −2 2i
1
1
i
i i
√ ,z=− √ ,z= √ ,z=− √
2 2
2 2
2 2
2 2
h
i
No solution
h
z=
√
√ i
2i, z = − 2i
h
√
√
√
√ i
z = (1 + 3)i, z = (1 − 3)i, z = −(1 + 3)i, z = −(1 − 3)i
z2 − |z2 |
= 0.
4 + |z2 |
3
Ex. 187. z(2 + |z2 |) = .
z̄
h
Ex. 186.
Im(z) = 0
i
h
i
|z| = 1
14
3 Limits of one real variable funtions
3.1 Check, using the definition, the following limits
Verify the definition of limit in the following cases:
Ex. 188. lim x = 1.
x→1
Ex. 189. lim
x→+∞
1
= 0.
x
Ex. 190. lim(2x + 1) = 7.
x→3
Ex. 191. lim x2 = 4.
x→2
1
= +∞.
x→0 x2
Ex. 192. lim
Ex. 193. lim
x→0
1
x3
@.
Ex. 194. lim 3x = 3.
x→1
Ex. 195. lim sin x = 1.
x→π/2
Ex. 196.
lim
x→(π/2)−
tan x = +∞.
Ex. 197. lim+ x − [x] = 0.
x→1
Ex. 198. lim− x − [x] = 1.
x→1
Ex. 199. lim+ log1/2 x = +∞.
x→0
x+2
1
= .
x→+∞ 2x + 2
2
Ex. 200. lim
Ex. 201. lim sin
x→+∞
1
= 0.
x
3.2 Computation of limits
Calculate, if they exist real or infinite,the following limits:
1
Ex. 202. lim x2 + .
x→2
x
15
h9i
2
A. Berretti, F. Ciolli
Exercises in Mathematical Analysis I
Ex. 203. lim
x + x2
.
x→+∞ x3 + 1
h i
0
x sin x
.
x→0 1 − cos x
√
Ex. 205. lim
x2 + 4 − x .
h i
2
Ex. 204. lim
h i
0
x→+∞
Ex. 206. lim
x→+∞
√
2x + x2 − x .
h i
1
log2 (x + x2 )
.
x→+∞ log x − 1
3
h
i
2 log2 3
Ex. 207. lim
sin x − x
.
x→0 x9/10
√
√
x2 + 1 − 17
Ex. 209. lim
.
x→4
x−4
h i
0
Ex. 208. lim
Ex. 210. lim+ 41/x .
h 4 i
√
17
h
i
+∞
Ex. 211. lim− 41/x .
h i
0
x→0
x→0
√
x
Ex. 212. lim
√ .
x→0 1 − cos 4 x
sin x −
h
sin x − x2
.
Ex. 213. lim √
x→0
1 − cos x2
h
4
−2
i
+∞
i
h i
1
Ex. 214. lim 2(sin x−1)/x .
x→π/2
sin2 x
+ cos x − 1
Ex. 215. lim 2
.
x→0
x2
x+1
Ex. 216. lim log4
.
x→+∞
x−1
h i
0
h i
0
log3 (x + 1)
.
x→0
x
h
log3 e
i
log1/2 cos x
h
log4 e
i
Ex. 217. lim
Ex. 218. lim
x→0
x2
.
Ex. 219. lim+ (sin x)1/ log2 x .
h i
0
x3
.
x→+∞ 2x
h i
0
log3 x
.
x→+∞
x
h i
0
x→1
Ex. 220. lim
Ex. 221. lim
16
A. Berretti, F. Ciolli
Ex. 222. lim
x→+∞
x3
2log3 (log2
Exercises in Mathematical Analysis I
h
.
x)
+∞
i
Determine domain and image of the following functions, indicating if they are periodic
and even or odd.
p
Ex. 223. f (x) = 2 sin2 x + cos x − 1.
Ex. 224. f (x) = log3 (sin3 x − cos3 x).
Ex. 225. f (x) = log1/2 (| sin 2x| + cos x).
Ex. 226. f (x) = 4(sin x+cos x)/(sin x−cos x) .
Ex. 227. f (x) =
1
.
2sin x − 3cos x
1
, al variare di α ∈ R.
x3
2 + ex
Ex. 229. f (x) = arcsin 2x
.
e −3
p
4
Ex. 230. f (x) = tan2 (x2 + 1) − tan(x2 + 1) − 6.
Ex. 228. f (x) = |x|α sin
Ex. 231. f (x) =
5x + 5−x
.
2
Ex. 232. f (x) = arctan
5x − 5−x
.
2
Draw a qualitative graph of the functions studied in the exercises 223, 226, 228, 231 and
232 above.
Calculate, if they exist real or infinite,the following limits:
r
x+1
Ex. 233. lim (x + 5)
− x.
x→+∞
x−1
h i
6
h i
1
Ex. 234. lim x(log(x + 1) − log x).
x→+∞
Ex. 235. lim
x→+∞
x3 − 2x + 1
x2 + x3
!(2x2 +1)/(x−3)
h i
e−2
.
log cos x
.
x→0 sin 2x2
h
Ex. 236. lim
2 +3x log x
Ex. 237. lim+ (sin x)x
x→0
.
log(1 + sin x)
.
x→0 sin 2x + x2 log x
−
1i
4
h i
1
h1i
Ex. 238. lim
2
17
A. Berretti, F. Ciolli
Exercises in Mathematical Analysis I
e2x−3 − e−3
.
x→0
sin x
√
sin 1 + x2 − 1
.
Ex. 240. lim
x→0
x
h
Ex. 239. lim
2e−3
i
h i
0
!2
log(1 + x) + sin x + x
.
Ex. 241. lim
x→0
x + x2
√ 2
e−1/x + log 1 + x1/5 − sin 3 x
Ex. 242. lim
.
√
√
3
x→0
x−2 5x
h i
9
h
−
sin(x5 /3x )
.
x→+∞
x4 2−x
1i
2
h i
0
Ex. 243. lim
Calculate, if they exist real or infinite,the following limits:
√
x − cos(x − 1)
Ex. 244. lim
.
x→1
log x
πx 1/ log(3−x)
Ex. 245. lim sin
.
x→2
4
h1i
2
h i
1
Ex. 246. lim |x − 1|x−1 .
h i
1
Ex. 247. lim+ x1/ log x .
hi
e
x→1
x→0
Ex. 248. lim
x→+∞
cos(1/x)
cos(2/x)
!(x2 +1)/x
h i
1
.
√
(2xx − 1)1/ x − 1
Ex. 249. lim+
.
√
x→0
x log x
h i
2
h
Ex. 250. lim x2 ((e1/x + 1)1/2 − cos(1/x)).
x→+∞
+∞
h3i
2
Ex. 251. lim x2 ((2e1/x − 1)1/2 − cos(1/x)).
2
x→+∞
√
2
e−1/(3−x) + e(4 − 3 cos(x − 3))1/5 − e
Ex. 252. lim
p
x→3
1 − cos(x − 3)
Ex. 253. lim sin(1/x) · log(x2 + e1/x + 2x
x→+∞



1
sin
Ex. 254. lim
10

x→+∞ log (x2 + x + 1) 
√
arcsin x
.
Ex. 255. lim q
√
x→0
4
cos x − 1
i
2 /(x+1)
4−x
h
.
).
1
x
log10 (x3 + x + 1)
x+1
−1


 .

e i
− √
2
h
i
log 2
h 3 i
( )10
2
h
18
−4
i
A. Berretti, F. Ciolli
Exercises in Mathematical Analysis I
hi
e
Ex. 256. lim(1 + sin x)1/ arctan x .
x→0
Calculate, if they exist real or infinite,the following limits:
x2 + sin x
.
2
x→+∞
x + log x + e2x
h1i
Ex. 257. lim
4 −x
Ex. 258. lim x e
x→+∞
√
Ex. 259. lim
x→+∞
2
+ sin(1/x ) + 1
2
√1+2x4
h √ i
e 2
.
√
!
4x
+
1
x + x3 − x log √
2 x+3
.
x arctan x
−
h
−∞
i
h
+∞
i
xarctan x − xπ/2
.
√
x→+∞
(1 + x)π/2+1/ log x
Ex. 260. lim
xarctan x − xπ/2
.
x→+∞ (1 + x)π/2−1
Ex. 261. lim
e−1/x + x2 +
Ex. 262. lim+
1
log2 x
+ x log e−1/x + e−2/x + 1
.
ex − 1
x→0
x sin x − cos x + ex /2
Ex. 263. lim+ √
.
x→0
1 − cos x · arcsin x
2
(x2 − 2x + 1) tan(x − 1) − sin3 (x − 1)
.
p
x→1
cos(x − 1) − 1
√
x cos x3 − 1 + sin2 x3/4
.
Ex. 265. lim+
√
√ 2
x→0
x3 e−1/ x + x ex − 1
√
x cos x3 − 1 + sin2 x3/4
2
√ .
Ex. 266. lim+
√
x→0
x3 e−1/ x + ex − 1 / x
h 4 i
√
2
h i
0
Ex. 264. lim
Ex. 267. lim+
x→0
3i
π
h i
0
h
log | log x| + log x
.
log 1 + xlog x
h
+∞
i
h i
1
h i
0
2
e3x−x − e2 cos(x − 1) − x + 1
Ex. 268. lim
.
x→1
log (sin πx/2)
h 2e i
( )2
π
Calculate, if they exist real or infinite,the following limits:
xx
x
h
i
1; 0; 1
Ex. 269. lim+ xx ; lim+ xx ; lim+ xx .
x→0
x→0
x→0
n times
z}|{
.x
..
Ex. 270. lim+ xx
x→0
h
i
1 if n is even, 0 if n is odd
.
19
A. Berretti, F. Ciolli
Ex. 271. lim(
x→1
Exercises in Mathematical Analysis I
(x − 1)2
)log x .
sin(πx)(e − ex )
h i
1
2
Ex. 272.
Ex. 273.
Ex. 274.
Ex. 275.
cos(1/x) − e−1/x
lim
.
r
x→+∞ √
2+2
x
x4 − x2 − x2 log
x2 + 1
p
2
sin(ex − cos x + 2 sin x2 1 + 2 sin2 x)
lim
.
x→0+
2 sin2 x
q
√
2
ex − cos x + 2 sin x3 1 + 2 sin x3
lim
.
x→0+
2 sin3 x
√
√
1 + x sin x − cos 2x
.
lim
x→0+
tan2 (x/2)
h
Ex. 276. lim
x→0
Ex. 277. lim
x→0
sin2 x2
√
log(cos2 x)(x − x2 + 3x + 1)
1+
e−1/x
− cos x
i
h7i
4
h
+∞
i
h i
6
h √e i
2
log(2 − cos x)(2 − cos x)1/x sin x2
−2
.
2
h i
2
.
( √1+3x2 −1)/x2
Ex. 278. lim (sin x + 2)2 log(sin x + 1)
.
h i
0
x→0
!
e−1/x
+ 1 + sin3 (1/x)
log
x4
!
Ex. 279. lim
.
x→+∞
2 + x3
log
x3
h1i
2
Arrange in growing order of infinity (infinitesimal) the following functions and sequences,
after having determined the order of infinite (infinitesimal), if it exits as a real number.
Ex. 280. For x → +∞: a)
ex
,
x2
x2
1
, d)
.
log x
sin(1/x)
n2
3
n
c) n , d)
.
2
b) x log x,
Ex. 281. For n → +∞: a) 2n ,
b) n!,
Ex. 282. For x → +∞: a) xx ,
b) x log2 x,
c)
c) x2 log x ,
√
3
1 − cos x
c) √
,
arcsin x
1
Ex. 283. For x → 0+ : a)
,
log x
b) x ,
Ex. 284. For x → 0+ : a) log x,
b) log | log x|,
2
20
c)
d)
h
i
d, b, c, a. ord d=1
h
i
a, d, b, c
x+1
x5 + x3 + 2
log
.
2
x +1
h x
i
b, d, c, a. ord d=2
d) log x · arcsin x.
h
i
a, c, d, b. ord b=2, c= 16
1
,
x log x
d)
1
.
log(1 + x)
h
i
b, a, c, d. ord d=1
A. Berretti, F. Ciolli
Exercises in Mathematical Analysis I
Ex. 285. For x → 1+ : a) e−1/(x−1) ,
2
Ex. 286. For x → 2+ : a)
√
10
b)
1
,
(x − 2)3/2
c) sin3
x − cos(x − 1),
x−1
x2 − x, d)
.
log20 (x − 1)
h
i
c, d, b, a. ord b=1, c= 13
√
1
,
3/4
(x − 2) log(x − 2)
b)
√
3
c) e
x−2/ sin(x−2)
,
h
i
a, b, c, d. ord a= 32
d) (x − 2)1/(2−x) .
Ex. 287. For x → 0+ : a) x arctan x,
1 − cos x
,
log x
b)
√
4
d) sin3 x.
i
h
d, c, b, a. ord a=2, d= 34
c) xx − 1,
Arrange in growing order of infinity (infinitesimal) the following functions and sequences,
after having determined the order of infinite (infinitesimal), if it exits as a real number.
!1+1/ √log x
2
2
x
x
3
Ex. 288. For x → +∞: a) x2 , b) log(1 + x3 + ex ), c)
, d)
.
x+1
x+1
h
i
c, d, a, b. ord a=2, b=3, c=1
√
Ex. 289. For n → +∞: a)
n
,
2
n +1
b)
1
,
n log n
x2 (1 − cos x)2
log(1 + sin4 x)
Ex. 292. For x → +∞: a)
log2 n
,
n
√
b) n( 3 + n2 − n),
√
Ex. 290. For n → +∞: a) ( n n − 1)−1 ,
Ex. 291. For x → 0+ : a)
c)
,
b) log(x+1),
x2 log(2 − cos(1/x))
sin2 (1/x)
√
,
x x
b) 100 ,
x
d) x log100 (1 + x).
(x−3)2
Ex. 293. For x → 3+ : a) (e (3−x)(x+1)3 − 1) sin(x − 3)9/4 ,
d) (x − 3)3 log10 (x − 3).
d)
n!
.
(n + 1)! − (n − 1)!
h
i
c, d, b, a. ord a= 32 , d=1
c) (cos(1/n) − 1) · 2n /(n+1) , d) nn .
h
i
a, b, c, d. ord b=2
3
c) x log x,
d) sin(x log(1+x))·log x.
h
i
c, b, d, a. ord a=2, b=1
!
x2 + 1
c) x log
,
x
h
i
d, a, c, b. ord a=2
2
b) sin3 (x − 3), c) (x − 3)3 log(x − 2),
h
i
d, b, a, c. ord a= 13
,
b=3,
c=4
4
 √
25
3
2+x 

x


Ex. 294. For x → 0+ : a) x log(1 + x2 ), b) x
, c)  √
 , d) x3 log10 x.
4
2
h x + 2x
i
25
b, c, d, a. ord a=3, b=2, c=4, c= 12
√
!
√
(1 − cos x)2 x + 1
x 2 + 1 √x
2
+
Ex. 295. For x → 0 : a) x arctan x, b) √
, c) x log
e ,
x
x4 + 1 log(1 + x2 )
h
i
d) sin(x3 log x).
a, c, b, d. ord a= 32 , b=2
2−x/(x2 +1)
21
A. Berretti, F. Ciolli
Exercises in Mathematical Analysis I
Calculate the limit of the following sequences:
2
en
Ex. 296. lim n .
n→+∞ n
h
+∞
i
3/2
en
Ex. 297. lim
.
n→+∞ nn2 + en
√
n2 + n3 − n + sin n
Ex. 298. lim
.
√
4
n→+∞
1 + n5 + 2n6
1/2
2(1+log
Ex. 299. lim
n→+∞
n1/2
n)
h i
0
√
h 4 2i
2
h i
0
.
√
Ex. 300. lim (log(n2 + 1) − log n − log(n + 1)) 1 + n2 .
h
n→+∞
2n − 3n
.
n→+∞
4n
−1
i
Ex. 301. lim
h i
0
Calculate the limit of the following sequences:
p
n
Ex. 302. lim (n2 + 1) sin(1/n).
h i
1
n→+∞
nn
.
n→+∞ (n!)!
√
n
Ex. 304. lim n!.
h i
0
Ex. 303. lim
h
n→+∞
+∞
i
√
Ex. 305. lim log
n→+∞
1−2 n+1
n n+ √n
.
n+1
(n−1)n n
n! (n+1)!
Ex. 306. lim 1 + n
.
n→+∞
n
√
2
Ex. 307. lim n + 1 arcsin e−n +
h i
1
h√ i
e
e
n→+∞
1
.
n2 + n
h i
0
n2
Ex. 308. lim (1 + cos(1/n) − cos(2/n))−(arcsin(1/n)) .
n→+∞
Ex. 309. lim ( √
n→+∞
n2 + n3 + 3
n + n + n3 − 1
h i
1
h i
1
e−1/n )n .
4
n6 + en log n + 2n arcsin(1/n)
Ex. 310. lim
.
3
n→+∞
nn − n! + en
p
n2
2
Ex. 311. lim
n3 + 1 + en .
h i
1
hi
e
n→+∞
Ex. 312. lim
n→+∞
Ex. 313. lim
n→+∞
en + sin(πn/2).
hi
e
2 + sin n.
h i
@
p
n
√
n
22
A. Berretti, F. Ciolli
Exercises in Mathematical Analysis I
*Ex. 314. Let {an } be a positive terms sequence such that
lim log
n→+∞
an
≥ 0.
an+1
Give at least two counterexamples showing that from this relation is not possible deduce that
lim an = +∞
n→+∞
Moreover, say under which further hypothesis the result would be true.
Ex. 315. Using the comparison theorem, show that
lim
n→+∞ n2
1
1
1
+ 2
+ ··· + 2
= 0.
+1 n +2
n +n
*Ex. 316. Let {an } be a positive terms sequence. Show that
an+1
=r≥0
n→+∞ an
lim
⇒
lim
n→+∞
√
n
an = r
Use the sequence an = e1/n + sin(πn/2) + 1 to show that in general the converse is not true.
*Ex. 317. Show, exhibiting a counterexample, that if {an } is a non-negative terms sequence,
then
lim a1/n
n =l
n→+∞
;
an
lim
n→+∞ ln
=1
Moreover, show that if lim a1/n
n = l > 1 then an → +∞ for n → +∞.
n→∞
23
4 Study of functions of one real variable
4.1 Asymptotes
Determine the possible asymptotes (vertical, horizontal, oblique) for the following functions,
after having indicated their domain. Moreover, calculate the limit of the functions to the
boundary points of their domain.
x+1
.
3 − 2x
1
.
Ex. 319. f (x) =
x(x − 2)
√
x4 + 1
.
Ex. 320. f (x) =
x−2
Ex. 318. f (x) =
Ex. 321. f (x) = x log(1 + x).
x
.
+1
x
Ex. 323. f (x) = 2
.
x −1
Ex. 322. f (x) =
x2
Ex. 324. f (x) = x arcsin
Ex. 325. f (x) = elog
2
1
.
x+1
(x/(x−1))+log(3x−3)+2 .
Ex. 326. f (x) = log(1 − 3ex + 2e2x ).
2
Ex. 327. f (x) = x ex/(x −1) .
r
x
.
Ex. 328. f (x) = x cos 2
x +1
Ex. 329. f (x) =
√
log |x|
+ x2 + 2x.
3 + log |x|
Ex. 330. f (x) = x arctan x.
(Use the formula arctan x + arctan
Ex. 331. f (x) = x1+1/ log x .
√
2
Ex. 332. f (x) = x1+log x/ 1+log x .
Ex. 333. f (x) =
x2 −1/x2
e
.
x4 − 1
24
1 π
= , x > 0).
x
2
A. Berretti, F. Ciolli
Exercises in Mathematical Analysis I
4.2 Continuity and derivability
Determine the domain and the set of continuity of the following functions.
Ex. 334.



 x − [x] − 1, x ≤ 2
f (x) = 

 x − [x],
x > 2.
√
Ex. 335. f (x) = [x] +
x − [x].
Ex. 336. f (x) = 41/ sin x .
Ex. 337. f (x) =
sin(log x)
.
log x
Ex. 338.



 sin(cot x), x , kπ, k ∈ Z
f (x) = 

 0,
x = kπ, k ∈ Z.
Ex. 339. Determine a ∈ R such that the following function result to be continuous
 2

x −1


, x , −1

f (x) = 
x+1


 a,
x = −1.
Ex. 340. Say if it is possible to apply the Weierstass theorem about the existence of the
extremes to the following function



0≤x<1
 x,
f (x) = 

 1 − x, 1 ≤ x ≤ 3.
Determine the set of continuity and the set of derivability of the following functions and
calculate their derivative.
Ex. 341. f (x) = tan 2x.
Ex. 342. f (x) = e2x − e−2x .
Ex. 343. f (x) = 32x .
2 +1
Ex. 344. f (x) = xx
.
2x + 3
.
x−4
r
2x + 3
Ex. 346. f (x) =
.
x−4
√
x
Ex. 347. f (x) = 2
.
x +1
Ex. 345. f (x) =
25
A. Berretti, F. Ciolli
Exercises in Mathematical Analysis I
Ex. 348. f (x) = (arcsin x)3 .
Ex. 349. f (x) = esin x .
x
.
Ex. 350. f (x) = arctan
1 − x2
Ex. 351. f (x) = log tan x.
Ex. 352. f (x) = arcsin
Ex. 353. f (x) = arcsin
1
1+
!
√
x
!
.
x2
.
x2 − 1
Ex. 354. f (x) = x e1/(1−x) .
Ex. 355. f (x) = 2arccos 3x .
Ex. 356. f (x) = log 2|x|.
Ex. 357. f (x) =
log x
.
3 − 2 log(2x)
Ex. 358. f (x) = |x|x + ex .
Ex. 359. f (x) =
Ex. 360. f (x) =
√
x2 + x4 arctan x.
√
1 − cos x.
s
!
x2
Ex. 361. f (x) = log 2
.
x −1
The same work (determination of continuity, derivability and calculation of the derivative)
is recommended also for the functions in the exercises in paragraphs 1.8 and 4.1.
4.3 Invertibility and derivative of the inverse function
Verify the invertibility of the following functions and determine the domain of derivability
of the respective inverse functions.
Ex. 362. f (x) = 2x + log x.
Ex. 363. f (x) = −x + e−2x .
Ex. 364. f (x) = x|x| + log(1 + x).
Ex. 365. f (x) = x + sin x.
26
A. Berretti, F. Ciolli
Exercises in Mathematical Analysis I
√
Ex. 366. f (x) = x |x| + arctan x.
Ex. 367. f (x) =
√
5
1 − x − cos x.
Ex. 368. For any of the following function f (x), determine:
π
π
f 0 (2), f 0 (1), f 0 (1 + log 2), f 0 ( + 1), f 0 (1 + ), f 0 (0).
2
4
Moreover, write the equation of the tangent line passing for the point indicated.
Ex. 369. Use the mean value theorem to show that
| sin x − sin y| ≤ |x − y|,
x, y ∈ R .
4.4 Critical points
Determine the possible critical points for the following functions.
x
.
+1
x
.
Ex. 371. f (x) = 2
x −1
Ex. 370. f (x) =
Ex. 372. f (x) =
x2
log x
.
x
Ex. 373. f (x) = xe−1/x .
√ Ex. 374. f (x) = x 1 +
1 .
log x Ex. 375. f (x) = x log x.
Ex. 376. f (x) = x3 + x2 − x.
−x(x + 1).
3
1
x
Ex. 378. f (x) = e
|x| + (3x − 8) .
2
2
Ex. 377. f (x) =
p
Ex. 379. f (x) = ((2 − x)6 )log |x−2| .
4.5 Derivability and Monotony
Determine the intervals of monotony for the functions in the paragraph 4.4.
27
A. Berretti, F. Ciolli
Exercises in Mathematical Analysis I
4.6 Taylor and Mac Laurin Polynomials
Determine the Mac Laurin polynomial of the following functions to the indicated order.
Ex. 380. f (x) = sin(x2 ), to the order 4.
√
Ex. 381. f (x) = 1 + 2x, to the order 3.
Ex. 382. f (x) = log(1 + x3 ),
Ex. 383. f (x) = sin2 x,
Ex. 384. f (x) = ex+1 ,
to the order 8.
to the order 4.
to the order 5.
Determine the Taylor polynomial, centered in x0 and to the indicated order, for the
following functions.
Ex. 385. f (x) = ex , x0 = 2, to the order 3.
Ex. 386. f (x) = cos x, x0 = 3, to the order 4.
Ex. 387. f (x) = log(1 + x), x0 = 2, to the order 3.
Ex. 388. Determine the Mac Laurin polynomial of order 4, for the function
f (x) = log(1 + x sin x) .
Determine the Mac Laurin polynomial of order 5, for the following functions.
Ex. 389. f (x) = (1 + x)ex .
Ex. 390. f (x) = x sin x + cos x.
Ex. 391. f (x) = sin x · log(1 + x).
4.7 Using Taylor polynomials for the calculation of limits
Calculate the following limits.
x3 1/(x+1)
e
− 1 − x.
Ex. 392. lim
x→+∞ x + 1
4
1
x sin x + cos x − ex
2
Ex. 393. lim+
.
x→0
x2 log(1 + x2 )
√
e−1/(x+1) + xx − x log x
Ex. 394. lim+
.
x→1
x log (x cos(x − 1)) 2
h
h
−
h
x5 + x2 log x
x→+∞
2 arctan x
2
3
6
x + x log
− x5
π
π
−
3i
2
25 i
24
+∞
i
hπi
Ex. 395. lim
4
28
A. Berretti, F. Ciolli
Exercises in Mathematical Analysis I
4.8 Uniform Continuity
Ex. 396. Verify, using the definition, that f (x) = x2 is not an uniform continuous function
over X = [1, +∞).
Ex. 397. Establish if f (x) =
domains:
arctan x
is a uniform continuous function over the following
x
D1 = (0, +∞); D2 = (1, +∞); D3 = [1, +∞); D4 = (−∞, −1) ∪ (2, +∞).
Ex. 398. Verify if f (x) = x − log x results to be a Lipschitz function over the domain
D = [1, +∞).
Verify if the following functions result to be uniformly continuous over their domain:
Ex. 399.


−1/|x| ,


xe
f (x) = 


0,
Ex. 400.
if x , 0,
if x = 0.




2 sin x + 1,
f (x) = 


log(e(2x + 1)),
if x < 0,
if x ≥ 0.
Ex. 401. f (x) = sin(esin x )
For any of the following functions determine a ∈ R such that they result to be continuous.
Then check if, for such an a, the functions result to be also uniformly continuous throughout
their domain of definition.
Ex. 402.
Ex. 403.
Ex. 404.


x


a(e − 1),
f (x) = 


e−x ,
if x < 1,
if x ≥ 1.

log x

1−x



 x +e ,



f (x) = 
a,



√
π


 2 − x + − arctan x,
4
√


2


 x − 2x + 2,
f (x) = 
a log(x + 1)



,

x
29
if x > 1,
if x = 1,
if x < 1.
if x ≤ 0,
if x > 0.
A. Berretti, F. Ciolli
Ex. 405.
Exercises in Mathematical Analysis I

1
2 sin x



,
−
p

 2

log(1
+
x)
+
1



f (x) = 


a,




x(ex + 1) − 1,
30
if x > 0,
if x = 0,
if x < 0.
5 Integrals of one-variable functions and numerical series
5.1 Immediate indefinite integrals (primitives)
Calculate the following indefinite integrals (primitives).
Z
1
Ex. 406.
dx.
√
4
x3
Z
p
Ex. 407.
3qx dx, q ∈ R+ .
Z
(a2/3 − x2/3 )3 dx,
Ex. 408.
Pn (x) dx,
Ex. 409.
Pn (x) =
n
X
h2
3
n
hX
i
ak k+1
x +c
k+1
ak x , ak ∈ R.
k
k=0
k=0
Ex. 410.
Z X
n
βk x
αk e
dx,
n
hX
αk
αk , βk ∈ R, βk , 0.
k=0
Ex. 411.
Z X
n
k=0
αk sin βk x dx,
αk , βk ∈ R, βk , 0.
h
−
k=0
x2 − 3x + 1
dx.
x
√
Z
3+ x
Ex. 413.
dx.
√
5
x2
√
Z
a + 1 − x2
Ex. 414.
dx,
√
1 − x2
Z
x2
Ex. 415.
dx.
1 + x2
Z
Ex. 416.
tan2 x dx.
n
X
α
k=0
Z
h1
Ex. 412.
2
Ex. 418.
βk
i
cos βk x + c
√
h √
i
10 10
5
5 x3 +
x11 + c
11
h
i
a arcsin x + x + c
a ∈ R.
h
i
x − arctan x + c
h
h
cot2 x dx.
Z
k
βk
i
eβk x + c
i
x2 − 3x + log |x| + c
Z
Ex. 417.
i
(3q)1/2 x3/2 + c
i
h
9
1
9
a2 − a4/3 x5/3 + a2/3 x7/3 − x3 + c
5
7
3
a ∈ R.
Z
h
i
4x1/4 + c
1 + 2x2
dx.
x2 (1 + x2 )
h
31
−
i
tan x − x + c
i
− cot x − x + c
i
1
+ arctan x + c
x
A. Berretti, F. Ciolli
Z
sin 2x
dx.
cos x
Z
x5 + 1
dx.
x+1
Ex. 419.
Ex. 420.
Z
Ex. 421.
dx
2
sin x cos2 x
Z
Ex. 423.
sin2
x
dx.
2
cos2
x
dx.
3
Z
Ex. 425.
Z
Ex. 426.
h x5
5
h xn
a ∈ R.
n
.
+a
−
i
− 2 cos x + c
i
x4 x3 x2
+
−
+x+c
4
3
2
i
xn−1
xn−2
+ a2
+ · · · + an−1 x + c
n−1
n−2
h
i
tan x − cot x + c
cos 2x
dx.
sin x + cos x
Z
Ex. 424.
h
xn − an
dx,
x−a
Z
Ex. 422.
Exercises in Mathematical Analysis I
h
i
cos x + sin x + c
h1
2
h1
i
(x − sin x) + c
i
3
2x
sin
+c
2
4
3
h
i
x
x
2 tan − 2 cot + c
2
2
1
x
x dx.
sin2 cos2
2
2
x+
5.2 Indefinite integrals by substitution
Calculate the following indefinite integrals using, for instance, the method of substitution of
variable.
Z
√
Z
Ex. 428.
Z
Ex. 429.
Ex. 431.
Ex. 432.
Ex. 433.
Ex. 434.
3
x
dx.
1 − x2
a ∈ R+ .
1
dx,
− x2
a ∈ R+ .
a2
i
sin3/2 x + c
i
1
log 1 − x2 + c
2
h1
i
x
arctan + c
a
a
h 1
i
a − x −
log +c
2a
a+x
h
1
dx,
a2 + x2
Z
Ex. 430.
h2
sin x cos x dx.
Ex. 427.
Z √
a−x
dx, a ∈ R+ .
√
a+x
Z
1 + e−x
dx.
1 + xe−x
Z
1
dx, a, b ∈ R+
√
2
a − bx
Z
x
dx.
√
1 − x2
−
h
i
x 1√ 2
a arcsin +
a − x2 + c
a 2
h
i
log |x + ex | + c
r
h 1
i
b
x+c
√ arcsin
a
b
h √
i
− 1 − x2 + c
32
A. Berretti, F. Ciolli
Z
Ex. 435.
√
1
x 5x − 7
Z
Ex. 436.
e−x
h 2
√ arctan
7
h 1
dx.
sinα x cos x dx,
Z
Ex. 437.
Exercises in Mathematical Analysis I
α , −1.
1
dx.
+ ex
cos(log x)
dx.
x
Z √
x
439.
dx.
1+x
Z
x2
440.
dx.
(x − 1)3
Z
x
441.
dx, a , 0.
√
a4 − x4
Z
cot x
442.
dx, α ∈ R+ .
sinα x
Z
1
443.
dx, a , 0.
√
x x2 − a2
Z √ 2
x − a2
444.
dx, a ∈ R.
x
Z
ax + b
445.
dx, a, b, c, d ∈ R, c , 0.
cx + d
Z
e2x
446.
dx.
√
ex − 1
Z
tan x
447.
dx.
log(cos x)
Z
1
448.
dx, a , 0.
(a + x)(a2 − x2 )1/2
Z
1
449.
dx.
sin x cos x
Z
1
450.
dx.
sin x
Z
1
451.
dx.
cos x
Z
1
452.
dx, a ∈ R+
√
x(a + x)
h
Ex. 438.
Ex.
Ex.
Ex.
Ex.
Ex.
Ex.
Ex.
Ex.
Ex.
Ex.
Ex.
Ex.
Ex.
i
5x + 7
+c
7
i
sinα+1 x + c
α+1
h
i
arctan ex + c
Z
Ex.
r
i
sin(log x) + c
h √
i
√
2( x − arctan x) + c
h
log |x − 1| −
i
2
1
+
c
−
x − 1 2(x − 1)2
h1
2
arcsin
i
x2
+
c
a2
i
1
+
c
α sinα x
h 1
i
a
− arctan + c
a
x
h
h√
−
i
a
+c
x
i
h1
a(cx
+
d)
+
(bc
−
ad)
log
|cx
+
d|
+
cons.
c2
h2 p
3
x2 − a2 − a arccos
i
√
(ex − 1)3 + 2 ex + 1 + c
h
i
− log log(cos x) + c
h
1
−
a
h
r
i
a−x
+c
a+x
i
log | tan x| + c
i
sin x
|+c
1 + cos x
h
i
1 + sin x
log |
|+c
cos x
r
h 2
i
x
+c
√ arctan
a
a
h
33
log |
A. Berretti, F. Ciolli
Z
Ex. 453.
√
Z
Ex. 454.
Z
Ex. 455.
Z
Ex. 456.
Z
*Ex. 457.
1
a2 + x2
Exercises in Mathematical Analysis I
h
a ∈ R.
dx,
1
dx,
2
(a + x2 )3/2
a , 0.
1
dx,
2
(a + x2 )5/2
a , 0.
1
dx,
2
(a + x2 )7/2
a , 0.
1
dx,
(a2 + x2 )(2n+1)/2
√
i
log a2 + x2 + x + c
h1
a2
√
x
a2 + x2
i
+c
i
x
x3
1
(
)
+
c
−
√
p
a4
a2 + x2 3 (a2 + x2 )3
h1
h1
a
(√
6
a , 0, n ∈ N.
x
−
a2 + x2
i
x3
x5
2
1
+ p
)+c
p
3 (a2 + x2 )3 5 (a2 + x2 )5
h
i
use the results of the previous exercises
5.3 Indefinite integrals by parts
Calculate the following indefinite integrals using, for instance, the method of integration by
parts.
Z
Ex. 458.
log x
dx.
x3
i
1
1
− 2 log x +
+c
2
2x
i
h 1
− sin2 x cos x + 2 cos x + c
3
h
Z
sin3 x dx.
Ex. 459.
Z
h3
sin4 x dx.
Ex. 460.
8
Z
h
sin5 x dx.
Ex. 461.
Z
Ex. 462.
sin x
dx.
ex
i
1
2
cos3 x − cos5 x + c
3
5
i
1
sin xe−x + cos xe−x + c
2
!
h x4
i
1
x3
arctan x +
x−
− arctan x + c
4
2
3
x3 arctan x dx.
h logα+1 x
Ex. 464.
i
1
1
sin2 x +
sin 4x + c
4
32
− cos x +
h
Z
Ex. 463.
x−
α+1
−
+ c if α , 0, 1; log |x| + c, if α = 0; log | log x| + c if α = −1
Z
h
i
xex − ex + c
xex dx.
Ex. 465.
Z
h
i
ex (x2 − 2x + 2) + c
x2 ex dx.
Ex. 466.
Z
xn ex dx,
Ex. 467.
n ∈ N.
h
i
ex (xn − nxn−1 + n(n − 1)xn−2 − · · · + (−1)n n!) + c
Z
Ex. 468.
i
h
x sin x dx.
34
i
− x cos x + sin x + c
A. Berretti, F. Ciolli
Exercises in Mathematical Analysis I
Z
h
x cos x dx.
Ex. 469.
Z
Ex. 470.
x2 sin x dx.
h
i
− x2 cos x + 2x sin x + 2 cos x + c
x2 cos x dx.
h
i
− x2 cos x + 2x sin x + 2 cos x + c
Z
Ex. 471.
"
Z
Ex. 472. In =
n
x cos x dx, In = −xn cos x + nIn−1 =
#
Z
n
n−1
n−2
= −x cos x + n(−x cos x + (n − 1)(−x cos x + . . . + 2 x cos x dx) . . .)
x sin x dx,
n ∈ N, n > 1.
Z
Let I1 =
"
Z
Ex. 473. In =
n
x sin x dx, In = xn sin x − nIn−1 =
#
Z
n
n−1
n−2
= x sin x − n(x sin x − (n − 1)(x sin x − . . . − 2 x sin x dx) . . .)
x cos x dx,
n ∈ N, n > 1.
Z
Let I1 =
!
i
x2 sin2 x
−x sin x cos x +
+c
+
2
2
2
Z
h1
2
x sin x dx.
Ex. 474.
Z √
h1 √
i
x 1 − x2 + arcsin x + c
2
1 − x2 dx.
Ex. 475.
h x2
Z
x arcsin x dx.
Ex. 476.
Z
Ex. 477.
2
x
dx.
cos2 x
arcsin2 x dx.
Z
2
Z
sin px cos qx dx,
Ex. 480.
p, q ∈ R, p , q.
h
Z √
x2 + a dx,
Ex. 481.
a ∈ R.
Z
ex sin x dx.
Ex. 482.
Z
ex cos x dx.
Ex. 483.
Z
Ex. 484.
Z
Ex. 485.
i
1 √
x 1 − x2 − arcsin x + c
4
h
i
x tan x + log | cos x| + c
h1
earcsin x dx.
Ex. 479.
arcsin x +
h
i
√
x arcsin2 x + 2 1 − x2 arcsin x − 2x + c
Z
Ex. 478.
i
x sin x + cos x + c
eαx sin x dx,
α ∈ R.
eαx cos x dx,
α ∈ R.
35
i
√
earcsin x x + 1 − x2 + c
i
p
q
sin
px
sin
qx
+
cos
px
cos
qx
+
c
q2 − p 2
q2 − p2
√
h1 √
i
(x x2 + a + a log x2 + a + x + c
2
h1
i
ex (sin x − cos x) + c
2
h1
i
ex (sin x + cos x) + c
2
h 1
i
αx
e
(α
sin
x
−
cos
x)
+
c
α2 + 1
h 1
i
αx
e
(sin
x
+
α
cos
x)
+
c
α2 + 1
A. Berretti, F. Ciolli
Z
Ex. 486.
Z
Ex. 487.
Exercises in Mathematical Analysis I
eαx sin βx dx,
(α, β) ∈ R2 , (α, β) , (0, 0).
h
i
1
αx
e
(α
sin
βx
−
β
cos
βx)
+
c
α2 + β2
eαx cos βx dx,
(α, β) ∈ R2 , (α, β) , (0, 0).
h
i
1
αx
e
(β
sin
βx
+
α
cos
βx)
+
c
α2 + β2
Z
ex cosn x dx,
*Ex. 488.
n ∈ N.
(Use the formula:

!
bn/2c


1 X n



cos(n − 2k)x,
n odd

n−1

k
2


k=0
cosn x = 
!

bn/2c−1
X n!


1
n
1


cos(n − 2k)x, n even
+ n−1


 2n n/2
k
2
k=0
and the result of the exercise 487.)
Z
*Ex. 489.
ex sinn x dx, n ∈ N.
(Use the formula:

!
bn/2c


1 X
n

bn/2−kc


sin(n − 2k)x,
(−1)

n−1

k
2


k=0
sinn x = 
!
!

bn/2c−1
X


1
n
1
n

bn/2−kc

+ n−1
(−1)
sin(n − 2k)x,


 2n n/2
k
2
n odd
n even
k=0
and the result of the exercise 486.)
Z
*Ex. 490. Im,n =
sinm x cosn x dx,
m, n ∈ Z.
(One obtains the following equivalent reduction formulas:
sinm−1 x cosn+1 x m − 1
sinm+1 x cosn−1 x n − 1
+
Im−2,n+2 =
+
Im+2,n−2 =
n+1
n+1
m+1
m+1
sinm+1 x cosn+1 x m + n + 2
sinm+1 x cosn+1 x m + n + 2
=
+
Im+2,n = −
+
Im,n+2 )
m+1
m+1
n+1
n+1
Z
*Ex. 491.
eαx sinm βx dx, α, β ∈ R, m ∈ Z.
Im,n = −
(Use the results of the previous exercises)
Z
*Ex. 492.
eαx cosn βx dx, α, β ∈ R, n ∈ Z.
(Use the results of the previous exercises)
5.4 Determine the following indefinite integrals (primitives).
Z
Ex. 493.
x2 + 2
dx.
(x − 3)2 (x + 2)
h 19
25
36
log |x − 3| −
i
11
6
+
log |x + 2| + c
5(x − 3) 25
A. Berretti, F. Ciolli
Z
Ex. 494.
Z
Ex. 495.
4x − 3
dx.
(x − 1)(x − 2)3
x5 + x4 − 8
dx.
x3 − 4x
Z
Ex. 496.
(x2
Z
Ex. 497.
Z
Ex. 498.
Exercises in Mathematical Analysis I
x
dx.
+ 1)(x − 1)
h
h x3
3
+
− log |x − 1| + log |x − 2| +
i
x2
+ 4x + 2 log |x| + 5 log |x − 2| − 3 log |x + 2| + c
2
h1 1
i
1
2
− log(x + 1) − arctan x + log |x − 1| + c
2 2
2
x+1
dx.
x2 + 1
x3 − 6
dx.
x4 + 6x2 + 8
i
2
5
−
+c
(x − 2) 2(x − 2)
h1
2
h
−
i
log(x2 + 1) + arctan x + c
i
5
1
+
+
log
|x
−
2|
−
log
|x
−
1|
+
c
2(x − 2)2 x − 2
Z
x3 − 2x2 + 5
dx.
x4 + 3x3 + 3x2 − 3x − 4
√
h1
i
1
5
31 7
2x + 3
2
log |x − 1| − log |x + 1| + log |x + 3x + 4| −
arctan √ + c
4
2
8
8
7
Z
2x3 − 3x + 3
dx.
(x − 1)(x2 − 2x + 5)
h 11
1
i
5
1
log |x2 − 2x + 5| + log (|x − 1|) − arctan (2x − 2) + 2x + c
4
2
2
4
Z
x2 + x + 1/2
dx.
x2 + 1
Ex. 499.
Ex. 500.
Ex. 501.
Z
Ex. 502.
h1
2
3x2 − 6x + 7
dx.
(x − 2)2 (x + 5)
h 16
7
log(x2 + 1) −
log (|x + 5|) +
i
1
arctan(x) + x + c
2
i
5
1
log (|x − 2|) −
+c
7
x−2
Z
2x2 + x
dx.
(x2 + 1)(x2 + 2x + 2)
h 1
1
i
1
− log |x2 + 2x + 2| + log(x2 + 1) + arctan (2x + 2) + c
2
2
2
Z
x3 + x − 1
dx.
(x2 + 2)2
Ex. 503.
Ex. 504.
h1
2
log(x2 + 2) −
i
x
x−2
arctan(
)
−
+
c
√
5
4x2 + 8
2
22
1
Z
1
dx.
+ 1)2
h1
2
i
2
1
x
log |x2 − x + 1| + log (|x + 1|) + 3 arctan(2x − √ ) + 3
+c
9
9
3x + 3
3
32
Z
h1
i
1
x
Ex. 506.
dx.
arctan
x
+
+
c
2
(x2 + 1)2
2x2 + 2
Z
4
Ex. 507.
dx.
4
x +1
√ !
√ !!
Ex. 505.
(x3
h
4
√
√
1
1
2x + 2
1
2x − 2
log |x2 + 2x + 1| − 5 log |x2 − 2x + 1| + 3 arctan
+ 3 arctan
√
√
2
2
2
22
22
22
1
5
2
37
i
+c
A. Berretti, F. Ciolli
Z
Ex. 508.
Exercises in Mathematical Analysis I
tan2 x
dx.
tan3 x + 1
h1
1
1
log | tan2 x − tan x + 1| + log (| tan x + 1|) − log tan2 x + 1 +
6
6
4
2
Z
Ex. 509.
sin x
cos2 x + 2 sin2 x
Z
Ex. 510.
h
dx.
x−
!
2 tan x − 1
√
i
1
3
− x+c
√
2
3
arctan
√
2 tan x
√
2
i
+c
1
dx,
m
sin x cosn x
m, n ∈ N.
hi
cos mx sin nx dx,
m, n ∈ N.
hi
Z
Ex. 511.
arctan
√
Z
x
dx.
x+1
4 1
5
1
4 40 1
3
1
2
1
i
h 1
x 4 + 1 − 20 x 4 + 1 − 4 log x 4 + 1 + c
20 x 4 + 1 + x 4 + 1 − 5 x 4 + 1 +
5
3
√
Z
3
x
Ex. 513.
dx.
√
x + x2
hi
Ex. 512.
√
4
Z
Ex. 514.
Z
Ex. 515.
1 + tan x
dx.
1 − tan x
1
dx.
3 + 5 cos x
h
1
2
i
log tan2 x + 1 − log (| tan x − 1|) + c
h 1
i
1
sin x
sin x
2 log + 2 − log − 2 + c
8
cos x + 1
8
cos x + 1
38
A. Berretti, F. Ciolli
Exercises in Mathematical Analysis I
5.5 Definite Integrals
Determine the following definite integrals:
Z 3
x
Ex. 516.
dx.
2
−2 x + 1
Z 3
x
Ex. 517.
dx.
2
−3 x + 1
Z 3
x2
dx.
Ex. 518.
2
−3 x + 1
Z 3
Ex. 519.
sin3 x cos x dx.
h log 2 i
2
h i
0
h
i
6 − 2 arctan 3
h i
0
−3
2π
Z
h i
0
sin3 cos 2x dx.
Ex. 520.
0
π/2
Z
Ex. 521.
x
sin2 x
π/4
hπ
dx.
4
+ log
π/2
Z
hπi
x sin x cos x dx.
Ex. 522.
√ i
2
4
−π/2
π
Z
Ex. 523.
h
x sin2 x dx.
0
Z
e
h e2
1 x| log x| dx.
e
Ex. 524.
4
5
e2x
dx.
√
2
ex − 1
Z 10
Ex. 526. Calculate
f (x) dx, where:
Z
h2 Ex. 525.
3


2

x ≤ −2,

x√ + 2 ,



2
 x −4
f (x) = 
, −2 < x < 2,



x


√

 x,
x ≥ 2.
10
f (x) dx,where:
Ex. 527. Calculate
−10


1


,
√

 x2 + 4
f (x) = 
 x2



,
 2
x +1
39
i
1
3 i
− 2
2 4e
i
√
√
(2 + e5 ) e5 − 1 − (2 + e2 ) e2 − 1
−10
Z
+
π2
4
x ≤ 0,
x > 0.
A. Berretti, F. Ciolli
Z
Exercises in Mathematical Analysis I
5
f (x) dx,where:
Ex. 528. Calcolate
−3


x



sin 2 ,
f (x) = 

x


cos ,
2
x > 0,
x < 0.
Ex. 529. Calcolate the area of the surface between the graph of the curves of equation
y = x3 , y = 2 − x2 under the condition x < 0. Say if it exists (finite) the area of the surface
between the graph of the two curves, without the condition x < 0.
Ex. 530. Calcolate the area of the surface between the graph of the curves of equation
y = −x2 + x + 2 ed y = x2 − 1.
40
A. Berretti, F. Ciolli
Exercises in Mathematical Analysis I
5.6 Improper Integrals
5.6.1 Determine the convergence or divergence for the following improper integrals
Z 1
h i
1
Ex. 531.
dx. Calculate, if it exist, the value of the integral.
π
√
−1
1 − x2
Z log 3
h
i
1
Ex. 532.
dx.
Divergent
ex − 3
0
Z ∞
1
dx, α ∈ R.
Ex. 533. I =
x logα x
2
h
i
1
If α > 1 , I =
;
if
α
≤
1
,
then
I
is
divergent.
(α − 1) logα−1 2
Z 6
h
i
1
dx.
Divergent
Ex. 534.
4 (x − 4) − log(x − 3)
Z 4
h
i
1
Ex. 535.
dx.
Convergent
3/5
2 |cos πx/2|
Z +∞
h
i
1
Ex. 536.
dx.
Divergent
5
1/5
((log x)(x + x − 2))
1
Z +∞
h
i
sin x log x
Ex. 537.
dx.
Convergent
(x + 1)3/2 − 1
0
Z +∞ 1/x2
h
i
e
− e1/x
Ex. 538.
dx.
Convergent
√
x
1
Z π/2 −1/x
h
i
e
Ex. 539.
dx.
Convergent
√
sin x
0
Z +∞
h
i
1
+
dx,
m
∈
R
.
Convergent
Ex. 540.
mx + ex
0
Z +∞
h
i
1
Ex. 541.
dx.
Convergent
p
2
(log x)2 (x3 + x)
5.6.2 Discuss the integrability in an improper sense of the following integrals.
Z +∞
h
i
log(t + 1)
Ex. 542.
dt.
Convergent
t3 + 2t + 1
1
Z 1
h
i
log t
Ex. 543.
dt.
Convergent
5/4 t1/2
0 (1 − t)
Z +∞
h
i
1
Ex. 544.
Divergent
√
√ dt.
t(t2 + 1) log(1 + t)
0
41
A. Berretti, F. Ciolli
Z
+∞
Ex. 545.
log(2 + x2 )
dx.
√
x arctan x2
+∞
e−y
1
Ex. 547.
+∞
Ex. 548.
1/2
Z
Ex. 549.
−1
+∞
Z
1/2
Z
+∞
Ex. 551.
1/2
Z
+∞
Ex. 552.
Ex. 553.
1
h
i
Convergent
dy.
e−x
dx.
(x − 3)1/3 (x − 1/2)1/2
h
i
Convergent
e−x
dx.
(x − 4)2 (x + 1/2)1/3
1
h
i
Divergent
dy.
h
i
Divergent
1
dx.
3/4
|x − 3| (x − 1/2)1/2
h
i
Convergent
(y −
3)1/3 (y
−
1/2)1/2
log(3 + x−1/4 )
(x − 3)3/4 (x − 1/2)1/2
3
Z
2y + arctan(y1/4 )
+∞
Ex. 550.
h
i
Divergent
2 /2
p
0
Z
h
i
Divergent
dy.
+∞
Ex. 546.
Z
1
sin √
y
(y − 1)1/2
0
Z
Exercises in Mathematical Analysis I
log x2
h
i
Convergent
dx.
h
i
Divergent
dx.
(1 − x)9/4 x1/2
Z +∞
h
i
e−x
Ex. 554. If Ia =
dx,
find
a
∈
R
such
that
I
<
+∞.
a
>
3
a
(x − 3)2 (x − 1/2)1/2
a
Z +∞
dy
Ex. 555. If Ia =
dy, find a ∈ R such that Ia < +∞.
(1 + y)2 (y + 2)a
1
h
1
2i
Moreover, calculate I1 .
a > −1. I1 = + log
2
3
0
5.6.3 Determine the values of α ∈ R s.t. the following improper integrals result to be
Ex.
Ex.
Ex.
Ex.
convergent.
Z 1
(tan x)α
556.
dx.
0 log(1 + sin x)
Z +∞
arctan(1/xα )
557.
dx.
√
x+2
0
Z 1
cos x + 3
558.
√ dx.
α
x
0 x +
Z +∞
arctan(x + 7)
559.
dx.
x logα (x − 2)
2
h
α>0
i
h
1i
α>
2
h
i
α<1
h
42
α>1
i
A. Berretti, F. Ciolli
Z
+∞
logα (1 + 1/x)
dx.
√
x+1
+∞
|sin(1/x) − 1/x|α /2
dx.
√
3
x
1 α
1 − cos 3 xα/2 dx.
x
Ex. 560.
2
Z
Ex. 561.
1
Z
+∞
Ex. 562.
1
Z
Exercises in Mathematical Analysis I
+∞
Ex. 563.
h
2
α>
1i
2
h
2i
|α| >
3
h
2i
α>
11
√
(arctan x)α ( x + 3)2α dx.
[Divergent for any α]
x2α + 1
√ ) dx.
x
[Divergent for any α]
0
Z
+∞
Ex. 564.
(e−x +
0
Z
+∞
Ex. 565.
−1
Z
+∞
Ex. 566.
arctan(x2 + 3)
dx.
(x + 1)α (x + 2)
h
arctan(1/x)α (x2 + 3)2α dx.
h
−
0
Z
+∞
Ex. 567.
3
Z
+∞
Ex. 568.
0
Z
2
Ex. 569.
−1
Z
0<α<1
i
1
<α<0
4
e−t
h
i
α<1
√ dt.
(t − 3)α t
√
sinα (1/ t)
dt.
√
t logα (t + 1)
[Divergent for any α]
(ex+3 + 7 sin2 x)
dx.
xα (ex + 1)
+∞
Ex. 570.
2 /2
e−αx
i
h
i
α<1
h
i
α>0
dx.
−∞
Z
+∞
Ex. 571.
1
Z
+∞
Ex. 572.
4
Z
+∞
Ex. 573.
0
Z
+∞
log(1 + xα )
dx.
x3
Ex. 575.
0
Ex. 576.
0
1
h
logα+1 (x − 3)
dx. Moreover, calculate its value for α = −1.
√
ex−4 − 1
x
sin 2
x +1
dx.
(x2 − sin x2 )α
3 + 2 sin x
dx.
(x − 1)1/3 (x + 2)4α
0
Z
log(2 + x)
dx.
x2
+∞
Ex. 574.
Z
(e1/x − 1)α
h
α > −1
i
3 i
α > − .π
2
h
1i
0<α<
3
h
α>
1i
6
h
i
α>2
1
dx.
α
x(− log x) + x2 (1 − x2 )1/3
h
i
α>1
43
A. Berretti, F. Ciolli
Exercises in Mathematical Analysis I
5.6.4 Determine the values of α and β such that the following improper integrals
converge
Z 1
| log x|α
dx.
Ex. 577.
β
0 | sin πx|
Z +∞ αx+β/x
e
dx.
Ex. 578.
x+1
0
Z +∞
(arctan x)α
Ex. 579.
dx.
xβ (2 + cos x)
0
h
i
β < 1, β − α < 1
h
α < 0, β ≤ 0
i
h
i
β > 1, β − α < 1
5.7 Numerical Series
5.7.1 Determine the nature of the following numerical series
Ex. 580.
∞
X
k=1
Ex. 581.
∞
X
k=1
Ex. 582.
1
√ .
k+ k
h
i
Divergent
k
.
k + log k
h
i
Divergent
∞
X
k=1
1
k
h
i
Convergent
.
log k
!k
∞
X
log(log k)
.
Ex. 583.
log k
h
i
Convergent
∞
X
(k!)2
.
(2k)!
h
i
Convergent
k=1
Ex. 584.
k=1
Ex. 585.
∞
X
√
k 2 e−
k
h
i
Convergent
.
k=1
Ex. 586.
∞ √
X
k=1
Ex. 587.
∞ √
X
1
k2 + 1 − k log 1 + .
k
k+1−
h
i
Convergent
√ 2
k .
h
i
Divergent
k=1
r

∞ 
X
 3

Ex. 588.
 1 + sin − 1 1 − e−1/k .
k
h
i
Convergent
∞
X
2
1
Ex. 589.
(e1/k − 2 cos + 1).
k
h
i
Convergent
k=1
k=1
44
A. Berretti, F. Ciolli
Ex. 590.
∞
X
k=1
Ex. 591.
∞
X
k=1
Exercises in Mathematical Analysis I
1
,
3 + eαk
α ∈ R.
h
i
Convergent if α > 0, divergent otherwise
k2
,
4 + eαk
∈ R.
h
i
Convergent if α > 0, divergent otherwise
∞ 
2
X
√ 
1 k 

Ex. 592.
.
1 − e cos
k 
h
i
Convergent
k=1
Ex. 593.
∞ X
k=1
Ex. 594.
∞
X
k
h
i
Convergent
.
!
3
α
.
log(1 + √
)− √
3
3
k2
k2
k=1
Ex. 595.
5
9 − 2 cos k
∞ X
k sin
k=1
h
i
Convergent if α = 3, divergent otherwise
k3
1
.
k
h
i
Convergent
5.7.2 Determine the nature of the following series
Ex. 596.
∞ X
1−
k=1
Ex. 597.
∞ X
k=1
1
k 2
k1/3
3
5 + cos2 k
h
i
Convergent
.
k
h
i
Convergent
.
!
∞
X
(−1)n+1
6
+
Ex. 598.
.
3n
4n
Moreover, if it exists, calculate the sum.
k=1
Ex. 599.
∞
X
n
√
e
n2
n=1
n
[Converges to
23
]
3
h
i
Convergent
.

!
∞ 
X
 3x2 − 3 2n

n
+
1

 ,
+ 2
Ex. 600.
 x2 + 1
n (log n)x + 2 
k=1
x ∈ R.
h
i
√
Converges if1 < x < 2, diverges otherwise
5.7.3 Determine the nature of the following series for any α ∈ R and x ∈ R
∞
α
X
h
i
1
1 k
1
−
.
Converges
for
any
α
∈
R
Ex. 601.
k
k2
k=4
Ex. 602.
∞
X
nα (x + 1)2n
n=1
(2n)!
,
h
i
Converges for any α, x ∈ R
x ∈ R.
45
A. Berretti, F. Ciolli
∞
X
Exercises in Mathematical Analysis I
1
Ex. 603.
n 1 − 1 + 2α
n
n=1
Ex. 604.
∞
X
n=1
Ex. 605.
∞
X
1/4 !
h
i
Converges if α > 1, divergent otherwise
.
n8
.
(n − log n)10 − nα
α
n
4
n − 5n
2 1/4
h
i
Converges if α , 10, divergent otherwise
3
− n − 3n
1/3 h
i
Convergent if α < 0, divergent otherwise
.
n=1
Ex. 606. Determine the values of α ∈ R such that the following two series have the same
∞ ∞
X
X
h
i
α
log(1 + nα ).
α ≥ −1
nature:
e(n +1/n) − 1 ,
n=1
n=1
Ex. 607. Study the nature of the series
∞
X
3(−1) 3αn for any α ∈ R. Moreover, calculate its
n
n=1
sum once calculated the one of the two series
∞
X
n=1
3·3
2αn
and
∞
X
1
n=1
3
· 3(2n+1)α .
h
Converges for α < 0 to the value
9 + 3α i
3(1 − 32α )
5.7.4 Discuss the simply and absolute convergence of the following series
Ex. 608.
∞
X
arctan
n=0
Ex. 609.
∞ X
n=0
Ex. 610.
1
.
n+1
α
2α + 3
h
i
Simply and absolutely convergent
n
1
.
log n
h
i
Simply convergent if α < −3, absolutely convergent if α < −3, α ≥ −1
∞
X
1/4
α
(−1)k e1/k − 1 .
h
i
Simply convergent if α > 0, absolutely convergent if α > 4
k=1
46