Types of Integration

Integration are of two types:- i) Indefinite ii) definite

The integration written in the form is called indefinite integral.

The integration written in the form is called definite integral.

In this chapter we only discuss the indefinite integrals. The definite integrals will be discussed in
the next chapter.

Algebra of Integrals

i. =

ii.

iii. )= f(x)

Simple Integration Formula of some standard functions

i) dx = kx + c

ii dx = + c, n -1

iii) dx =ln +c

iv)

vi)

vii

viii

ix)

xi

xii)

xiii
xiv xv)

xvi) xvii)

Methods of integration

1. Integration by using standard formula.

1. INTEGRATION BY USING FORMULAS:-

Example -1 Evaluate the following

(i)

Ans :-

= 5 dx+5 { by algebra of integration}

= 5x

= 5x

(ii)

Ans :-

=3

= 3x

= 3x

=
 (iii) dx

Ans :- dx

= 4

= 4 +

(iv)

Ans :-

= 6x

(v)

= =

= 5

= 5tanx 5x + c

(vi)

Ans :-

= = { 1- cosx = 2 }

= +c

(vii) dx

Ans :- dx

= =

= = dx xdx
 = .

(viii)

Ans :-

= +

(ix) }dx

Ans :- }dx ( )

= =

= ( )

(x)

Ans:-

= = { =1}

= = tanx secx + c

(xi)

Ans:- = { we know here ae is in place of a}

= +c= + c (Ans)
(xii)

Ans: - =

= dx

= sinx + c

2. INTEGRATION BY SUBSTITUTION:-

When the integral cannot be determined by the standard formulae then we may
reduce it to another form by another variable t (as
) which can be integrated easily. This is called substitution method.

wher .

The substitution depends upon the nature of the given integral and has to be properly
chosen so that integration is easier after substitution. The following types of substitution are very
often used in Integrations.

TYPE I

TYPE II

Put

=>

=
TYPE III

Put f(x)=t

Differentiate both sites w.r.t

=>

TYPE-IV

Put f(x)=t

=>

SOME USE FULL RESULTS

1.

Ans :- Put ax+b = t

Differentiate both sites w.r.t x,

a=

=> dx =

=
 2.

Ans:-

Put

Differentiate both sites w.r.t x,

=>

3.

Ans :-

= dx (multiply & divide by sec x)

Put secx = t

Differentiate both sites w.r.t x.

sec x tan x =

=>

4.

Ans :-

=
Divide numerator & denominator by

Let tan

=>

=>

Now tan = = = = = cosecx cotx

Hence = ln I cosecx-cotx I + c

5.

Ans:-

= ( )

As tan( ) = secx+ tanx ( we can easily verify it by applying trigonometric formulae.

Hence = ln I secx+tanx I + c
BY APPLYING ABOVE FORMULA WE OBTAIN FOLLOWING

1.

Proof : -

Put ax+b=

Differentiate both sites w.r.t x.

a=

Similarly we can get the following results.

2.

3.

4.

5.

6.

7.
 8. =

9.

10.

11. =- +c

. = +c

The above results of substitution may be used directly to solve different integration problem.

Example 2 integrate the following

(i)

Ans :- { Let then =>

= = = = (ans)

(ii)

Ans :-

Let

Differentiate both sides w.r.t x

=>

=>

Now =

= .

(iii)

Ans:

Put 3x+5 =t

Differentiate w.r.t x

= =

= =

(iv)

Ans :-

Put

Differentiate w.r.t x

=> ( )

=> 5( ) dx =

=>

dx =

=
 (v)

Ans :-

Put

Differentiating both sides w.r.t x

=>

dx =

(vi)

Ans :-

Put

Differentiating both sides w.r.t x,

=>

= =

(vii)

Ans :-

Put =>

viii) Evaluate

Ans:- ( Let t = => dt =

= = =
ix) Integrate (2015-S)

Ans:- ( Let 2 - 5x = t => - 5 dx = dt => dx = - )

= = .

x) Evaluate (2019-W)

Ans:- ( Put )

INTEGRATION OF SOME TRIGONOMETRIC FUNCTIONS

If the integrand is of the form sinmx cosnx ,sinmx sinnx or cosmx cosnx,a trigonometric
transformation will help to reduce.it to the sum of sines or cosines of multiple angles which can
be easily integrated.

sinmxcosnx =

sinmx sinnx=

cosmx cosnx=

Example 3

i)Evaluate

Ans :-

= +c

=
ii)Evaluate

Ans :-

= =

iii)Evaluate

Ans :-

iv)Evaluate

Ans :-

= ( )

=
v)Evaluate

( cos3x = 4 => )

vi)Evaluate

Ans :-

= =

{ Put => cosx= => }

vii)Evaluate

Ans :-

{ Put sinx = => cosx = =>d = cosxdx}

viii)Evaluate

Ans :-

=> c

= =

= - =

INTEGRATION BY TRIGONOMETRIC SUBSTITUTION

TRIGONOMETRIC INDENTITIES
1- = )

, can be simplified by putting

X=a sin
X=a tan
X=a sec
X=a cos
X=a cot
X=a cosec
 Note

1.The integrand of the form a2 - x2 can be simplify by putting x= a sin (or x = a cos )

2. The integrand of the form x2+a2 can be simplify by putting x= a tan (or x = a cot )

3.The integrand of the form x2 - a2 can be simplify by putting x= a se (or x = a cosec )

Example -4

i) Integrate

Ans :-

Let x=a sin

Differentiate both sites w.r.t x

dx = a cos d

And x = a sin =>

ii)Integrate

Ans :-

Let

diffentiating both sides w.r.t x,

And

=
 =

= = +c

iii) Integrate

Ans :-

Diffentiating w.r.t x we have,

Hence = = =

= ( )

(x = a tan => )

= +c

= +c

= ( where k = c- ln|a| is a constant)

Ans :-

Let =>

= =

{ As => tan = = }

= =

Hence

v)Integrate

Let x = => and

Now =

= =

= =
 =

vi)Integrate

Ans :-

=>

Now

= =

{ As =

=> }

=
 =

= (

vii)Integrate

Ans :-

=>

= =

= =

{ As

=
 =

These 7 results deduced in Example-4 are sometimes used to find the integration of some other
functions. Some examples are given below.

Example-5 :-Integrate

Ans :- (As )

= (using formula )

Example 6: - Integrate

Ans :-

= { Let => }

Now

{ , here a= 3}

Example 7:- Integrate

Ans :- ( multiplying numerator and denominator by )

= ( Let => )
 =

= ( using formula = , here a=2 )

Example 8 ; -

Integrate

Ans :-

= { Let x+3 = t =>dx = dt }

= ..........................(1)

( putting => 2t dt = dz => t dt = )

= + = =

= ......................(2)

= ( applying formula = , where a=4 )

=2

= .................(3)

From (1),(2) and (3) we have,

= ( where is a constant)
ILATE Rule:
The choice of 1st function is made based on the order ILATE. The meaning of these letters is given below:

I – Inverse trigonometric function,

T – Trigonometric function

E – Exponential function

Table 1 gives a proper choice of 1st and 2nd functions in certain cases. Here 𝑚, 𝑛 may be zero or any
positive integer.
Table 1:

Function 2nd Choice 1st Choice

Integrate

Ans :- { from table-1, 1st function = cosx and 2nd function = x }

= {

Example

Integrate

Ans:- and 2nd

= { again by parts is applied taking as 1st and x as 2nd function.}

= = (

Example:

Integrate

Ans :-

{ There is no direct formula for and two functions are not multiplied with each other in
this integral. This type of integration can be solved by using integration by parts by writing
as 1.

= =(

{ Let

=
Example:

Integrate

Ans : -

=(

Example:

Integrate

Ans :-

Example

Evaluate

Ans:-

=
 =

( choosing )

=> + =

Hence =

Example:

Integrate

Ans:- =

(taking

= -

= .

In some cases, integrating by parts we get a multiple of the original integral on the right hand
side, which can be transferred and added to the given integral on the left hand side . After that
we can evaluate these integrals. Some examples of such integrals are given below.

Example18: -

Integrate

Ans:-L

SCTE&VT LEARNING MATERRIAL ON ENGINEERING MATHEMATICS-II

2.

3.

4.

5.
Integration by Partial Fractions Formula
The list of formulas used to decompose the given improper rational functions is given below. Using
these expressions, we can quickly write the integrand as a sum of proper rational functions.