Calculus II Final Exam Review – University Level – Comprehensive Summary of Techniques, Integrals, and Series Tests

1 Techniques of integration


a substitution
use when the integral contains a function
and its derivative
Example

1 1 2 1 dx
Ix city
dx fu du
4 C
x2 1 c
f
raytigf.ee


I 2 0
312413

C Partial fractions
examp

I
5 12 Ax B x 5 1 2 4

15 12 Ax 5Ax BX 5 B Cx 4c

15 12 Ax x2 Bx 5Ax 4C 513

5 12 Atc B 5A 41 5 B

A 3

41 5B 12 5B 4 A 12


IEE.IE
a
II
5

, Iiii s 4 Is
PracticeProblem

Six In ax si 2 3 1
In 21 Fenixtil c

e In 2s In

EE ii i
75 3 a
3
3331113

A
9 13
d Integration by Parts
formula
fudu uv fudu
Practice problems
fxetdx

Y
xe
ax

f e dx
xe etc

trigonometric Integrals
Theseinvolveintegrals thatinclude sinx cosx tan x seex


ee
E.EE Ii
cos2x 1 osczx
2

, use identities to reduce even powers
If powers are odd take oneout and use
identities to simplify

Example sins cosdx fu du
c sin c
u sinx
du costdx

PracticeProblem sin cos x dx 1 10312x 1 x
02
1 2x 1 1 02147
q 4
1 103 12 cost dx Cos 4
4

fx cos tx dx

f sin ax c



Trig substitution
use for integralsinvolving squareroots

Fx asino
2 Ear x a sect

3 Tax a ton

Example

Ex a

2sint dx 2costdo
costdo
teasing
cos to
a e
sin't cos'o 1
cos'o 1 sin't
50do I do C
21
ÉE 5 E C

, IE bTE
PracticeProblem

ax
Sx a

ÉÉsÉÉantdo
3sec tent do
19sec't
vÉoa
3sec tone do
9sec o É
psecto sent
Betendo
do cos do
age

g
sint c
Ff c


3sec
V4
sect

cos
it
a

Improper Integrals


Linef
2
f dx x
time x 1

converges
o
2 ax timef.tt timeIn

diverges