Topics for 115 Exam 2
The exam covers sections 7.2-7.5, strategies for integration, and 7.7. In general, the exam covers
techniques of integration, approximating definite integrals, and using error bounds. You need to
be comfortable using u-substitution, integration by parts, trig identities, partial fractions, and trig
substitutions. Given an integral, you need to be able to determine which methods to use. You may
need to use more than one method on a given integral. You also need to know how to estimate the
value of a definite integral using the Midpoint, Trapezoidal, or Simpsons rule (you do not need to
memorize these rules, they will be given to you on the exam). Don’t forget to practice problems
that involve functions given by a table of values. You should also know how to estimate the error
in using one of these rules or figure out how big n needs to be to get a certain error (again the
formulas will be given to you, but you must know how to use them and how to find k). Below are
some formulas you should know. Memorize these!
Integration formulas
Z
xn+1
n
x dx =
+ C (n 6= −1)
n+1
Z
csc2 x dx = − cot x + C
Z
Z
1
dx = ln |x| + C
x
sec2 x dx = tan x + C
Z
csc x cot x dx = − csc x + C
Z
Z
sec x dx = ln | sec x + tan x| + C
sec x tan x dx = sec x + C
Z
Z
1
dx = tan−1 x + C
2
1+x
1
sin(kx) dx = − cos(kx) + C
k
tan x dx = ln | sec x| + C
Z
1
cos(kx) dx = sin(kx) + C
k
Z
Z
1
1
dx = ln |kx + m| + C
kx + m
k
Z
ekx dx =
ekx
+C
k
Some derivatives you should make sure you know:
1
d
[sin x] = cos x
dx
d
[cos x] = − sin x
dx
d
[tan x] = sec2 x
dx
d
[sec x] = sec x tan x
dx
d
1
[ln x] =
dx
x
Table of Trig Substitutions (These will NOT be given to you on the exam.
You must learn them. The identities sin2 x+cos2 x = 1 and tan2 x+1 = sec2 x,
and the half angle identities, however, will be given to you.)
Expression
a2 − x 2
a2 + x2
x2 − a2
Substitution
x = a sin θ
x = a tan θ
x = a sec θ
Identity
1 − sin2 θ = cos2 θ
1 + tan2 θ = sec2 θ
sec2 θ − 1 = tan2 θ
Here is what will be given to you on the front page of your exam (so you don’t need to memorize
these):
• Trig Identities:
sin2 θ = 12 (1 − cos(2θ))
cos2 θ = 21 (1 + cos(2θ))
sin(2θ) = 2 sin(θ) cos(θ)
sin2 (θ) + cos2 (θ) = 1
tan2 (θ) + 1 = sec2 (θ)
• Numerical Integration Formulas:
Rb
Simpson’s Rule a f (x) dx ≈ ∆x
3 [f (x0 ) + 4f (x1 ) + 2f (x2 ) + · · · + 4f (xn−1 ) + f (xn )]
5
with error bound |ES | ≤ k(b−a)
, where k ≥ |f (4) (x)| for a ≤ x ≤ b.
180n4
Rb
Trapezoidal Rule a f (x) dx ≈ ∆x
2 [f (x0 ) + 2f (x1 ) + 2f (x2 ) + · · · + 2f (xn−1 ) + f (xn )]
3
with error bound |ET | ≤ k(b−a)
, where k ≥ |f 00 (x)| for a ≤ x ≤ b.
12n2
Rb
Midpoint Rule a f (x) dx ≈ ∆x[f (x1 ) + f (x2 ) + · · · + f (xn )] where xi is the midpoint of
[xi−1 , xi ]
3
with error bound |EM | ≤ k(b−a)
, where k ≥ |f 00 (x)| for a ≤ x ≤ b
24n2
.
2
To study:
• Go over all examples from class
• Go over all of your Webwork
• Look over the suggested problems
Practice Problems:
1. Evaluate the following integrals
Z
x3
√
(a)
dx
1 + x2
Z
x
√
dx
(b)
1 + x2
Z
2x3 + 3x
(c)
dx
x2 + 1
Z
(d)
sin(4x) cos5 (4x) dx
Z
ln(x)
(e)
dx
x3
Z
(f)
sin2 x dx
Z
(g)
tan2 x dx
Z
(h)
sin5 x cos2 x dx
Z
(i)
x2 e−x dx
Z
(j)
tan3 x dx
Z
dx
√
(k)
x x2 − 9
Z
1
(l)
dx
2
2x + x
2. Evaluate the following integrals
Z
√
(a)
sin( x) dx
Z
1
√
(b)
dx
x− x+2
Z
(c)
x5 sec2 (x3 ) dx
3
Z
(d)
ln(x2 + 1) dx
3. Write out the partial fraction decomposition for
x2 + 2x + 9
. You do not need
x3 (x2 + 5)2 (x + 1)(x − 1)
to find the values of A, B, C, etc.
4. Write out the partial fraction decomposition for
2x + 4
and find the values of the
(x − 2)(x2 + 4)
constants A, B, etc.
Z
3
1
dx. We can approximate this integral using Simpson’s rule. How
1 x
large should n be to guarantee that the error in our approximation is at most .000001?
5. Consider the integral
Answers:
√
( 1 + x2 )3 p
1. (a)
− 1 + x2 + C
3
p
(b) 1 + x2 + C
(c) x2 + 21 ln |x2 + 1| + C
6
(d) − cos24(4x) + C
x
(e) − ln
−
2x2
(f)
(g)
(h)
(i)
(j)
(k)
1
4x2
+C
sin(2x)
+C
4
tan x − x + C
cos3 x 2 cos5 x cos7 x
−
+
−
+C
3
5
7
−x2 e−x − 2xe−x − 2e−x + C
tan2 x
− ln | sec x| + C
2
1
−1
3 cos (3/x) + C
1
2x
−
(l) ln |x| − ln |2x + 1| + C
√
√
√
2. (a) −2 x cos( x) + 2 sin( x) + C
√
√
(b) 4/3 ln | x + 2 − 2| + 2/3 ln | x + 2 + 1| + C
x3 tan(x3 ) 1
(c)
− ln | sec(x3 )| + C
3
3
(d) x ln(x2 + 1) − 2x + 2 arctan(x) + C
3.
A
B
C
Dx + E
Fx + G
H
J
+ 2+ 3+ 2
+ 2
+
+
2
x
x
x
x +5
(x + 5)
x+1 x−1
4.
1
x
− 2
x−2 x +4
5. 46
4