Chapter 3Techniquesof Integration

3.1 Integration of Rational Functions

A function of the form P(x)/Q(x), where P and Q are polynomials is called rational function.We have already considered integrals of simple rational functions. Now we shall consider in details about integration of rational functions.

3.1.1 Integration of the form (p,q are constants)To evaluate this integral determine M and N such that

Eequate the coefficients of like powers of x and solve the resulting simultaneous equations.Then evaluate the integrals as follows:

The second integral can be evaluated by completing the square. This is explained through an example.

Example :Evaluate Solution: Let us take

Equating the coefficients of x and independent tems, we have

Thus

3.1.2 Integration of the formusing partial fractions.It is said to be proper fraction if the degree of P(x) is less than degree of Q(x), otherwise, it is called improper fraction. An improper numerical fraction such as 7/3 may be written as 7/3= 2 + 1/3. Similarly, an improper fraction may be expressed, by division, as the sum of a polynomial and a proper fraction.

For example,

Proper fractions of the form N(x)/D(x) can often be expressed as the sum of simple fractionswhich than can be integrated

Rules of Partial FactionsHere we shall state the rules by which partial fractions may be found.

1.For each non-repeated linear factor (x- a) of D(x) include a fraction of the form where A is a non-zero constant.

2.For each non-repeated irreducible (i.e. which cannot be factorized) quadratic factor of D(x) include a fraction of the form where AndB are constants.

3.If D(x) contains the repeated factor , the corresponding partial fractions are

4.If D(x) contains the repeated factor , the corresponding partial fractions are

First write down the partial fractions according to the above rules and then clear the fraction by multiplying through by the denominator. The resulting expression is an identity, i.e. it is true for all values of x. To find the values of the constants two methods are available. first some selected values of x (usually zeros of the denominator) will often give some of the constants immediately. Second, coefficients of like powers of x may be equated and the resulting simultaneous equations solved. In some cases a mixture of these two method is convenient.

Example: Evaluate Solution: The integral can be expressed as

The denominator factorizes into . Now consider the partial fractions of the proper fraction part as follows:

Clearing the fractions give

Setting x = 0 yield . Equating the coefficients x2 and x, we obtain

which yields

Thus

The integral becomes

Exercise 3.1:1.Evaluate the following:

(a)(b) (c)

(d) (e)(f)

(g)(h)(i)

(j)(k)(l)

(m)(o)(p) 3.2 Integration of Trigonometric Functions:

Trigonometric Identities:

1. 2.

3. 4.

Some Standard Trigonometric Integrals

1.

3. 2.

4.

The above results may be verified by reversing the derivative formula.

Example: Evaluate This integral can be evaluated as follows:

Thus

Similarly

Exercise 3.2.1

1. Evaluate the following:

(a) (b)(c)

(d)(e) (f)

3.2.2 Integration of the form

Example :Evaluate

Solution:I =

Let,

Then, I =

Exercise 3.2.2:1. Evaluate the following:

(a)(b) (c)

(d) (e) (f)

3.2.3Integrals of the form

(i)

(ii)

(iii)

Example :Evaluate

Solution:I =Exercise 3.2.3:1. Evaluate the following:

(a) (b)

(c) (d)

3.2.4Integrals of the form

Example :Evaluate

Solution:I =

Let,

Then, I =

Exercise 3.2.4:

1. Evaluate the followings:

(a)(b) (c)

(e)(f)(g)

(h)(i)(j)

3.4 Integration using Trigonometric Substitution

Expression inthe integrandSubstitution

Table: 3.2

Note thatanintegration of the formcan be evaluated using any one of the above substitution.

Example1 :Evaluate

Solution: Let,

and

So,

Example2 :Evaluate

Solution:I =

Let,

So, I =

3

Exercise 3.4: 1. Evaluate the following:

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

3.5Integrals of the formInstruction:Write down

and then divide numerator and denominator by to convert a tangent function.The method is explained through an example.

Example: Evaluate

Solution:I=

[dividingumerator and denominator by ]

Let,

So, I =

Exercise 3.5:

1. Evaluate the followings:

(a)(b) (c)

(d) (e) (f)

1