INTEGRATION I L-19 I MATHS CLASS 12 INCERT/JEE(MAIN & Advanced)CBSE/State Board

Integration is a fundamental concept in calculus that is the reverse process of differentiation. While differentiation is used to find the rate of change, integration is used to find the total or accumulated quantity like area, volume, or displacement.

🔹 Types of Integration
Indefinite Integration

Represents a family of functions.

No limits (bounds) involved.

General form:

∫
𝑓
(
𝑥
)
 
𝑑
𝑥
=
𝐹
(
𝑥
)
+
𝐶
∫f(x)dx=F(x)+C
where
𝐹
′
(
𝑥
)
=
𝑓
(
𝑥
)
F
′
(x)=f(x), and
𝐶
C is the constant of integration.

Definite Integration

Represents a specific numerical value (area under a curve).

Has upper and lower limits.

General form:

∫
𝑎
𝑏
𝑓
(
𝑥
)
 
𝑑
𝑥
∫
a
b
​
f(x)dx
🔹 Basic Integration Formulas
Some commonly used formulas include:

∫
𝑥
𝑛
𝑑
𝑥
=
𝑥
𝑛
+
1
𝑛
+
1
+
𝐶
(
𝑛
≠
−
1
)
∫x
n
dx=
n+1
x
n+1

​
+C(n

=−1)
∫
1
𝑥
𝑑
𝑥
=
ln
⁡
∣
𝑥
∣
+
𝐶
∫
x
1
​
dx=ln∣x∣+C
∫
𝑒
𝑥
𝑑
𝑥
=
𝑒
𝑥
+
𝐶
∫e
x
dx=e
x
+C
∫
sin
⁡
𝑥
𝑑
𝑥
=
−
cos
⁡
𝑥
+
𝐶
∫sinxdx=−cosx+C
∫
cos
⁡
𝑥
𝑑
𝑥
=
sin
⁡
𝑥
+
𝐶
∫cosxdx=sinx+C
🔹 Methods of Integration
Substitution Method – Used when integrand becomes simpler with a change of variable.

Integration by Parts – Useful for products of functions.

Partial Fractions – Used when integrand is a rational function.

Trigonometric Identities – To simplify trigonometric integrals.

🔹 Applications of Integration
Finding area under curves

Calculating volume of solids of revolution

Finding displacement from velocity

Solving problems in physics, economics, and engineering

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