This document is from a unit on indefinite integration. It defines indefinite integration and antiderivatives. It provides rules for integration, including algebra rules and standard integrals involving trigonometric, exponential, logarithmic and algebraic functions. It describes methods for integration like integration by substitution, integration by parts, and integrating multiple trigonometric functions. It provides examples demonstrating these integration techniques and exercises for students to practice indefinite integration.

1. Indefinite Integration
Course- Diploma
Semester-II
Subject- Advanced Mathematics
Unit- IV
RAI UNIVERSITY, AHMEDABAD

2. Unit-IV Indefinite Integration
RAI UNIVERSITY, AHMEDABAD
4. Indefinite Integration
4.1 Definition— A function ( ) is an anti-derivative or integral of a function ( ) if
( )
= ( )
For all in the domain of . The set of all anti-derivative of f is the indefinite integral of with
respect to , denoted by
( ) = ( )
The symbol ∫ is an integral sign. The function f is the integrand of the integral and x is the
variable of integration.
4.2 Algebra on integration—
1.∫[ ( ) + ( )] = ∫ ( ) + ∫ ( )
2. ∫ ( ) = ∫ ( ) , for constant.
4.3 Standard integrals—
1.∫ = + ( ≠ − )
2.∫ = | | +
3.∫ = − +
4. ∫ = +
5. ∫ = | | +
6. ∫ = | | +
7.∫ = +
8.∫ = − +
9. ∫ = +
10.∫ = − +
11.∫ = +
12.∫ = +

3. Unit-IV Indefinite Integration
RAI UNIVERSITY, AHMEDABAD
13.∫ = + − +
14.∫ = + +
15. ∫ = + − +
16. ∫ = +
17. ∫ = +
18. ∫ = +
19. ∫ = + √ + +
20. ∫ = + √ − +
21. ∫ √ + = √ + + + √ + +
22. ∫ √ − = √ − + + √ − +
23. ∫ √ − = √ − + +
Example— Solve the integral ∫ .
Solution— ∫ = + = +
Example— Solve the integral ∫ .
Solution— ∫ = + = + = +
Example— Solve the integral ∫ − + .
Solution— ∫ 4 − 3 +
√

4. Unit-IV Indefinite Integration
RAI UNIVERSITY, AHMEDABAD
= 4 ∫ cos − 3 ∫ + 2 ∫ √
= 4 sin − 3 + 2 +
4.4 Methods of integration
4.4.1 Integration by substitution—
If integrand is not in the standard form and can be transformed to integral form by a suitable
substitution then such process of integration is called as integration by substitution.
Take these steps to evaluate the integral ∫ { ( )} ′( ) , where and ′ are contineous
function—
Step 1: Substitute = ( ) and = ′( ) to obtain the integral ∫ ( ) .
Step 2: Integrate with respect to .
Step 3: Replace by ( ) in the result.
Example—Solve the following integral ∫ .
Solution— ∫
= ∫ 2
= ∫ .
= 2
⟹ = 2
⟹ =
= ∫ .
= +
= 2 +
Example— Evaluate ∫( + ) .
Solution— ∫(2 + 3)
=∫ Let 2 + 3 =
= ∫ 2 =
= + =
=
( )
+
Example— Integrate ∫ .
Solution— ∫
=∫ Let =

5. Unit-IV Indefinite Integration
RAI UNIVERSITY, AHMEDABAD
= ∫ =
= ∫ ( )
=
= tan + =
= tan +
4.4.2 Integration by method of parts—
If and are functions of then the integral of product of these functions is given by—
= −
This rule is called as ‘integration by method of parts’.
The choice of first and second integral is given by—
I— Inverse
L—Logarithmic
A—Algebraic
T—Trigonometric
E—Exponential
The term appearing first in this series have to take first integral (u).
Example—Integrate ∫ .
Solution— ∫ cos
=( ∫ cos ) − ∫ (∫ cos )
= − ∫ sin = sin + cos +
4.4.3 Integration of terms involving multiple trigonometric functions—
1. cos = . 2 sin cos = [sin( + ) + cos( − ) ]
2. sin sin = [cos( − ) − cos( + ) ]
3. cos cos = [cos( − ) − cos( + ) ]
3. sin = (1 − cos 2 )
4. cos = (1 + 2 )
5. sin = (3 sin − sin 3 )
6. cos = (3 cos + cos 3 )
Example— Evaluate ∫ .
Solution—∫ sin 3 cos 2
= ∫ 2 sin 3 cos 2
= ∫(sin 5 + sin )

6. Unit-IV Indefinite Integration
RAI UNIVERSITY, AHMEDABAD
= − 5 − + = (cos 5 + 5 cos ) +
Example— Integrate ∫ .
Solution— ∫ sin
= ∫(3 sin − sin 3 )
= −3cos + cos 3 +
= cos + cos ) +
EXERCISE
Question— Integrate ∫ .
Question— Integrate ∫ sin 2 cos 3 .
Question— Integrate∫ sin 2 .
Question— Integrate∫ .
Question— Evaluate∫(3 − 5) .

7. Unit-IV Indefinite Integration
RAI UNIVERSITY, AHMEDABAD
References—
1. mathworld.wolfram.com
2. tutorial.math.lamar.edu
3. https://www.khanacademy.org
4. www.webmath.com
5. en.wikibooks.org/wiki/Calculus/Indefinite_integral