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

•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 given integrals using appropriate substitutions.

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 z

1 tan
2 x

From the identity cos2 y

2cos 2 y

1 if we let y
 x
2 then cos 2

2cos 2 cos x

2cos
2

1

1
 sec 2
2

2
 2

1

1 cos x

2
2 then simplifying we get

1

2 z 2


1
1
 z 2 z
2

From the identity sin 2 y
 y y if we let y
 x
2 then by doing the same steps done in cosine 2y we get since z
 x tan
2 dz
 sec
2 x
 
 dx
2 2
2 dz
2 dz



1 tan 2
 z 2
 dx
 

 dx dx

2 dz
1
 z 2 sin x

2 z
1
 z 2

EXAMPLES: Evaluate each of the following
 3 dx
5

4 cos x
 dx sin 2 x

4
 dx
 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 .
EXAMPLE: Evaluate
 dx x
 x

C. Algebraic substitution
I. Linear Function
If the integrand involves ( ax
 b ) n m
.
The substitution z
 n ax
 b will eliminate the radical.
EXAMPLE: Evaluate each of the following:
 x x

4 dx  x 5
 x

1

2 dx
 2 dx
1
 3 x

2

II. Quadratic Function
If the integrand involves and an odd power of x .
x
2  a
2 , x
2  a
2 ,
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 a
2  x
2
EXAMPLE: Evaluate
 x
3 x
2 
9 dx

,
D. Reciprocal substitution
If the integrand has a radical which cannot make use of the previous substitution methods, try:
Let x

1 z differentiate such that dx
  dz z
2
EXAMPLE: Evaluate
 dx x x 2 
2 x

1


Homework 2-4:
Evaluate each of the following:
 dx x 1
  x 2

1
4 dx x

2
4
 x 2 dx x
 2 dx
3

5 sin x
 dx x
 3 x

0

2 dx
5 sin x

3
 tdt
2 t

7
 3 dx sin x
 tan x


1
0 x 2 x

1 dx

7 x

2 x

1

3
2 dx
 x
2 
2 x x

1
 dx cos x
 sin x

1
  
2
 2
3 dt
 x x

4 dx
 cos xdx
3 cos x

5