Comprehensive Treatise on
Integration
For Class 12 and Engineering Level
Contents
1 Introduction to Integration 2
2 Standard Integration Formulas 2
3 Techniques of Integration 3
3.1 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3.2 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3.3 Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
4 Solved Examples 4
5 Graphical Illustration 4
6 Applications of Integration 4
6.1 Area under Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
6.2 Volume of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
6.3 Physics Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
6.4 Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
6.5 Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1
, Mathematics Integration Treatise
1 Introduction to Integration
Definition
Integration is the reverse process of differentiation. It is used to calculate areas, vol-
umes, total accumulated change, and more.
R
Notation: f (x) dx
Types of Integrals:
• Indefinite Integrals: General form without limits.
• Definite Integrals: Calculated over a closed interval [a, b].
2 Standard Integration Formulas
Algebraic and Exponential
xn+1
Z
xn dx = +C (n ̸= −1)
n+1
Z
1
dx = ln |x| + C
x
Z
ex dx = ex + C
ax
Z
ax dx = +C
ln a
Basic Trigonometric Integrals
Z
sin x dx = − cos x + C
Z
cos x dx = sin x + C
Z
sec2 x dx = tan x + C
Z
csc2 x dx = − cot x + C
Z
sec x tan x dx = sec x + C
Z
csc x cot x dx = − csc x + C
Page 2
Integration
For Class 12 and Engineering Level
Contents
1 Introduction to Integration 2
2 Standard Integration Formulas 2
3 Techniques of Integration 3
3.1 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3.2 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3.3 Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
4 Solved Examples 4
5 Graphical Illustration 4
6 Applications of Integration 4
6.1 Area under Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
6.2 Volume of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
6.3 Physics Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
6.4 Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
6.5 Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1
, Mathematics Integration Treatise
1 Introduction to Integration
Definition
Integration is the reverse process of differentiation. It is used to calculate areas, vol-
umes, total accumulated change, and more.
R
Notation: f (x) dx
Types of Integrals:
• Indefinite Integrals: General form without limits.
• Definite Integrals: Calculated over a closed interval [a, b].
2 Standard Integration Formulas
Algebraic and Exponential
xn+1
Z
xn dx = +C (n ̸= −1)
n+1
Z
1
dx = ln |x| + C
x
Z
ex dx = ex + C
ax
Z
ax dx = +C
ln a
Basic Trigonometric Integrals
Z
sin x dx = − cos x + C
Z
cos x dx = sin x + C
Z
sec2 x dx = tan x + C
Z
csc2 x dx = − cot x + C
Z
sec x tan x dx = sec x + C
Z
csc x cot x dx = − csc x + C
Page 2
