Techniques of Integration in Calculus: A Comprehensive Guide, Slides of Differential and Integral Calculus

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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

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 given integrals using appropriate

substitutions.

Integration by miscellaneous

substitution :

A. Integration of rational functions of sine and

cosine using half angle substitution

In this lesson we shall introduce several

substitution method to simplify the form of the

integrand. They are as follows:

B. Fractional powers of x

D. Reciprocal substitution

C. Algebraic substitution

From the identity cos 2^ y^ ^ 2cos^2 y ^1

if we let (^2)

x y  (^) then 2

2 2 2 2 2 2

cos 2 2 2cos 2 1

cos 2cos 2 1 (^2 ) sec (^2) (^2 ) 1 tan (^2) (^2 ) 1 2 1 1

                    (^)        (^)     (^)      (^)  

 ^  

x x

x^ x

x

x

z z z

2 2

1 z cos x 1 z

   then simplifying we get

From the identity

if we let 2

x y  then by doing the same steps done in cosine 2y we get

sin2 y  2sin y cos y

2 sin 2 1

 

x z z

since

 

2

2

2

tan 2 sec 2 2

2 1 tan 2 2 1



      

      ^        

z^ x

dz^ x^ dx

dz x dx

dz z dx^22 1

dz dx z

B. Fractional powers of x

If an integrand is a fractional power of the variable x

integrand can be simplified by the substitution^ x^  zn

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

EXAMPLE: Evaluate each of the following:

( ) n.

m ax  b

The substitution (^) z  n ax  b will eliminate the radical.

x^3 x  4 dx 

3

2 1 2

dx   x 

  x^5 x  1 2 dx 

D. Reciprocal substitution

If the integrand has a radical which cannot make use of the previous substitution methods, try:

x^1 z Let^ 

,

differentiate such that (^2) dz dx z

EXAMPLE: Evaluate 2

2 1

dx  x x  x 

Homework 2-4:

Evaluate each of the following:

3 5 sin

dx

 (^)  x cos sin 1

dx  x  x 

3 sin tan

dx

 x  x

cos 3 cos 5

xdx 0 2 ^ x^  5 sin 3

dx x

  

 x^ x^ ^4 dx

3

dx  x  x

2 7

tdt  t 

(^2 )

1

x x x

  

 

(^3 ) ^7 x^^2 x^ ^1 dx

t t  2 23 dt

(^0 )

  1 x^ x^ ^1 dx

4 (^1 )

dx  x 

4 x^2

dx

x



dx x  x  x 