Download Lesson 13 Integration by miscellaneous substitution

TOPIC
TECHNIQUES OF INTEGRATION
TECHNIQUES OF INTEGRATION
1. Integration by parts
2. Integration by trigonometric substitution
3. Integration by miscellaneous substitution
4. Integration by partial fraction
TECHNIQUES OF INTEGRATION
3. Integration by miscellaneous substitution
OBJECTIVES
•translate a rational function of sine and cosine
into a rational function of another variable;
•use the basic identities in evaluating integrals
involving rational functions of sine and cosine;
and
•evaluate the
substitutions.
given
integrals
using
appropriate
Integration by miscellaneous
substitution:
In this lesson we shall introduce several
substitution method to simplify the form of the
integrand. They are as follows:
A. Integration of rational functions of sine and
cosine using half angle substitution
B. Fractional powers of x
C. Algebraic substitution
D. Reciprocal substitution
,
A. Integration of rational functions of sine and cosine
Half-Angle Substitution
If an integrand is a rational function of sin x, cos x
and other functions, it can be reduced to a rational
function of z, by the substitution
1
z  tan x
2
From the identity cos2y  2cos y  1
2
x
if we let y  2
1  z2
cos x 
1  z2
x
x
cos 2    2cos2    1
2
2
x
cos x  2cos2    1
2
2

1
x
 
sec 2  
2
2

1
x
1  tan2  
2
2

1
2
1 z
2  1 z2

get
1 z2
then
then simplifying we
From the identity sin2y  2sin y cos y
x
if we let y 
2
x
since z  tan 2
2  x  dx
dz  sec   
2 2

2  x 
2dz  1  tan    dx
 2 



2dz  1  z dx
2
then by doing the same
steps done in cosine 2y
we get
sin x 
2dz
dx 
1 z2
2z
1 z2
EXAMPLES: Evaluate each of the following
3dx
 5  4 cos x
dx
 sin2x  4
dx
 1  sin x  cos x
B. Fractional powers of x
If an integrand is a fractional power of the variable x
integrand can be simplified by the substitution x  z n
where n is the common denominator of the exponents
of x .
dx
EXAMPLE: Evaluate
x
x
C. Algebraic substitution
I. Linear Function
m
n
If the integrand involves (ax  b) .
The substitution
z  n ax  b
will eliminate the radical.
EXAMPLE: Evaluate each of the following:
3
x
 x  4dx
 1
2dx
3
x 2
 x  x  1 dx
5
2
II. Quadratic Function
If the integrand involves
x a
2
2
,
x 2  a2
,
and an odd power of x.
Let u be the radical and perform the following:
1. Square both sides
2. Solve for x 2
3. Differentiate both sides to obtain x dx
EXAMPLE: Evaluate
3
2
x
x
 9 dx

a2  x 2
,
D. Reciprocal substitution
If the integrand has a radical which cannot make
use of the previous substitution methods, try:
Let
1
x
z
dz
differentiate such that dx   2
z
EXAMPLE: Evaluate
x
dx
x 2  2x  1
Homework 2-4:
Evaluate each of the following:
x

4
1

dx
dx
1 x  x 2
dx
x 2
4x
dx
x
2
2dx
 3  5 sin x
x

0
1


x
2
0

2
3
x
x  1dx
dx
5 sin x  3
tdt
2t  7
3
t
t

2
dt



3dx
 sin x  tan x
2
x
2
7
x
2
x

1
dx



3

x 2  2x
x  4dx
cos xdx
 3 cos x  5
x 1
dx
 cos x  sin x  1