Math 202
Assignment 5: The Fundamental Theorem of Calculus
The due date for this assignment is ..................
Reading assignment: Section 5:3
..................................................................................................................................................
1. Use Part 1 of the Fundamental Theorem of Calculus to Önd the derivative of the
function
Z ex
(a) h (x) =
ln (t) dt
1
Z 1
u3
(b) y =
du
2
1!3x 1 + u
2. Evaluate the integral.
Z 2
(a)
(1 + 2y)2 dy
1
Z 1
(b)
(xe + ex ) dx
0
Z 1
(c)
eu+1 du
Z!1&
(d)
f (x) dx where
0
f (x) =
&
sin x
cos x
if 0 & x < .n2
if .n2 & x & .
3. What is wrong with the equation?
Z 1
x!3 1
3
!
x!4 dx =
]!2 = "
"3
8
!2
4. Find the derivative of the function
Z sin x
! y=
ln (1 + 2v) dv
cos x
5
Math 202
Assignment 6: IndeÖnite Integrals and The Net
Change
The due date for this assignment is ..................
Reading assignment: Section 5:4
..................................................................................................................................................
1. Find the general indeÖnite integral.
Z
(a)
(1 + tan2 4) d4
2. Evaluate the integral.
(a)
Z
1
(x10 + 10x ) dx
0
(b)
(c)
Z
0
p
Z
0
(d)
(e)
&n4
Z
Z
1 + cos2 5
d5
cos2 5
3n2
p
p
1n 3 2
0
2
!1
dr
1 " r2
t "1
dt
t4 " 1
j x " 2 j x j j dx
6
Math 202
Assignment 7: The Substitution Rule
The due date for this assignment is ..................
Reading assignment: Section 5:5
..................................................................................................................................................
1. Evaluate the indeÖnite integral.
Z
(a)
5t sin (5t ) dt
Z
cos 5
(b)
d5
sin2 5
Z
sin 2x
(c)
dx
1 + cos2 x
Z
sin x
(d)
dx
1 + cos2 x
Z
dx
p
(e)
1 " x2 sin!1 x
Z
x
(f)
dx
1 + x4
Z
1+x
(g)
dx
1 + x2
2. Evaluate the deÖnite integral.
(a)
Z
&n4
(x3 + x4 tan x) dx
!&n4
(b)
(c)
Z
Z
2
1
1
0
p
x x " 1 dx
dx
p 4
(1 + x)
7
Math 202
Assignment 8: Integration by Parts
The due date for this assignment is ..................
Reading assignment: Section 7:1
..................................................................................................................................................
1. Evaluate the integral.
Z
(a)
(x2 + 2x) cos x dx
Z
p
(b)
ln ( 3 x) dx
Z
(c)
x 2x dx
(d)
(e)
(f)
(g)
Z
Z
Z0
Z
x e2x
dx
(1 + 2x)2
1
y
dy
e2y
cos x ln (sin x) dx
2
x4 (ln x)2 dx
1
2. First make a substitution and then use intgration by parts to evaluate the integral.
Z
p
(a)
cos x dx
(b)
(c)
Z
Z
p
p
&
&n2
' (
53 cos 52 d5
x ln (1 + x) dx
8
Math 202
Assignment 9: Trigonometric Integrals
The due date for this assignment is ..................
Reading assignment: Section 7:2
..................................................................................................................................................
1. Evaluate the integral.
Z &
(a)
cos4 (2x) dx
0
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
Z
&n2
sin2 x cos2 x dx
Z0
t sin2 t dt
Z
cos2 x tan3 x dx
Z
Z
cos5 x
p
dx
sin x
tan5 x dx
Z
x sec x tan x dx
Z
&n2
cot5 7 csc3 7 d7
&n4
Z
csc x dx
Z
&n6 p
1 + cos 2x dx
0
(k)
Z
x tan2 x dx
9
Math 202
Assignment 10: Trigonometric Substitution
The due date for this assignment is ..................
Reading assignment: Section 7:3
..................................................................................................................................................
1. Evaluate the integral.
Z p 2
x "9
(a)
dx
x3
Z p
1 + x2
(b)
dx
x
Z 0:6
x2
p
(c)
dx
9 " 25x2
0
Z
x
p
(d)
dx
x2 + x + 1
Z
p
(e)
x2 + 2x dx
Z
p
(f)
x 1 " x4 dx
10
Math 202
Assignment 11: Integration of Rational Function by
Partial Fractions
The due date for this assignment is ..................
Reading assignment: Section 7:4
..................................................................................................................................................
1. Evaluate the integral.
Z
4x
(a)
dx
3
2
x +x +x+1
Z 3
x + x2 + 2x + 1
(b)
dx
(x2 + 1) (x2 + 2)
Z
x+4
(c)
dx
2
x + 2x + 5
Z
1
(d)
dx
x3 " 1
2. Make a substitution to express the integrand as a rational function and then evaluate
the integral.
Z p
x+1
(a)
dx
x
Z
dx
p
(b)
2
x +x x
Z
x3
p
(c)
dx
3
2+1
x
Z
p
1
p
p
(d)
dx [Hint: Substitute u = 6 x]
3
x" x
Z
e2x
(e)
dx
e2x + 3ex + 2
Z
sec2 x
(f)
dx
tan2 x + 3 tan x + 2
Z
dx
(g)
1 + ex
11