What is the main difference between definite and indefinite integrals?

Indefinite → gives a family of functions with +C. Definite → gives a numerical value (area under curve).

Why is +C added in indefinite integration?

Because differentiation of a constant is zero, the original function can differ by a constant.

Can definite integrals be negative?

Yes, if the function lies below the x-axis, the integral is negative.

Which methods are most important for JEE?

Substitution, parts, partial fractions, and properties of definite integrals.

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Definite and Indefinite Integration

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Definite and Indefinite Integration

1.0Introduction to Integration

Integration is the inverse process of differentiation.

If: $\frac{d y}{d x} = f (x)$ , then: $y = \int f (x) d x$

Integration is classified into two types:

Indefinite Integration → Represents a family of functions (with constant of integration).
Definite Integration → Represents a numerical value, usually area under a curve within limits.

2.0Indefinite Integration

Definition

The indefinite integral of a function f(x) is written as: $\int f (x) d x = F (x) + C$

where:

F′(x)=f(x)
C = constant of integration

Properties of Indefinite Integration

Linearity: $\int [a f (x) + b g (x)], d x = a \int f (x), d x + b \int g (x), d x$
Reversal of Differentiation: $\frac{d}{d x} (\int f (x), d x) = f (x)$
Constant Factor Rule: $\int a \cdot f (x), d x = a \int f (x), d x$

Standard Formulas

$\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C (n \neq = - 1)$
$\int \frac{1}{x} d x = ln ∣ x ∣ + C$
$\int e^{x} d x = e^{x} + C$
$\int a^{x} d x = \frac{a ^{x}}{l n a} + C$
$\int sin x d x = - cos x + C$
$\int cos x d x = sin x + C$
$\int sec^{2} x d x = tan x + C$
$\int csc^{2} x d x = - cot x + C$
$\int sec x tan x d x = sec x + C$
$\int csc x cot x d x = - csc x + C$

Rules of Integration

Linearity: $\int [a f (x) + b g (x)] d x = a \int f (x) d x + b \int g (x) d x$
Substitution Rule: If x=ϕ(t), then: $\int f (x) d x = \int f (ϕ (t)) ϕ^{'} (t) d t$
Integration by Parts: $\int u v d x = u \int v d x - \int (\frac{d u}{d x} \int v d x) d x$
Partial Fractions: Used when integrand is a rational function.

Methods of Indefinite Integration

Substitution Method

If ( f(x) ) is complex, substitute ( x ) with another variable ( t ) to simplify the integral.

Example:

$\int 2 x cos (x^{2}), d x$

Let $(t = x^{2} \Rightarrow d t = 2 x d x)$

$\int cos t, d t = sin t + C = sin (x^{2}) + C$

Integration by Parts

Used when integrating the product of two functions: [ $\int u, d v = uv - \int v, d u$ ] Choose ( u ) and ( dv ) wisely (using ILATE rule: Inverse, Log, Algebraic, Trig, Exponential).

Partial Fractions

Decompose rational functions into simpler fractions before integrating.

Example: $\int \frac{1}{x ^{2} - 1}, d x = \int (\frac{1}{2 ( x - 1 )} - \frac{1}{2 ( x + 1 )}) d x$

Solved Examples on Indefinite Integration

Example 1: $\int (3 x^{2} + 2 x + 1) d x = x^{3} + x^{2} + x + C$

Example 2: $\int \frac{1}{x ^{2} + 1} d x = tan^{- 1} x + C$

Example 3 (By Parts): $\int x e^{x} d x = x e^{x} - \int e^{x} d x = (x - 1) e^{x} + C$

Explore in Detail: Indefinite Integration

3.0Definite Integration

Definition of Definite Integration

The definite integral of f(x) between a and b is defined as: $\int_{a}^{b} f (x) d x = F (b) - F (a)$ , where F(x) is the antiderivative of f(x).

Properties of Definite Integrals

Additivity over Intervals: $\int_{a}^{b} f (x), d x + \int_{b}^{c} f (x), d x = \int_{a}^{c} f (x), d x$
Reversal of Limits: $\int_{a}^{b} f (x), d x = - \int_{b}^{a} f (x), d x$
Zero Interval: $\int_{a}^{a} f (x), d x = 0$
Linearity: $\int_{a}^{b} [a f (x) + b g (x)], d x = a \int_{a}^{b} f (x), d x + b \int_{a}^{b} g (x), d x$
Symmetry: For even and odd functions over ([-a, a]):

Even: ( $\int - a^{a} f (x), d x = 2 \int_{0}^{a} f (x), d x$ )
Odd: ( $\int_{- a}^{a} f (x), d x = 0$ )

Evaluation Techniques

Direct Substitution: Find the antiderivative, then substitute the limits.
Change of Variable: If substitution is used, adjust the limits accordingly.
Splitting the Integral: Break into simpler intervals if necessary.

Application of Definite Integrals

Area under Curves: The area between the curve ( y = f(x) ), the ( x )-axis, and vertical lines ( x = a ) and ( x = b ) is: $A = \int_{a}^{b} f (x), d x$
Area between Curves: For two curves ( y = f(x) ) and ( y = g(x) ), where ( $f (x) \geq g (x)$ ) in ( [a, b] ): $A = \int_{a}^{b} [f (x) - g (x)], d x$
Other Applications

Physics: Calculation of displacement, work done, etc.
Probability: Continuous probability distributions.
Average Value: $Average value of f (x) over [a, b] = \frac{1}{b - a} \int_{a}^{b} f (x), d x$

Solved Examples on Definite Integrals

Example 1: $\int_{0}^{1} x^{2} d x = [\frac{x ^{3}}{3}]_{0}^{1} = \frac{1}{3}$

Example 2 (Even Function): $\int_{- 2}^{2} x^{2} d x = 2 \int_{0}^{2} x^{2} d x = 2 [\frac{x ^{3}}{3}]_{0}^{2} = \frac{16}{3}$

Example 3 (Substitution): $\int_{0}^{π /2} sin x d x = [- cos x]_{0}^{π /2} = 1$

Example 4 (Property): $\int_{0}^{1} ln (\frac{1}{x}) d x = \int_{0}^{1} ln (\frac{1}{1 - x}) d x$

Learn in detail : Definite Integration

4.0Difference Between Definite and Indefinite Integration

Feature	Indefinite Integration	Definite Integration
Definition	The process of finding the antiderivative (the inverse of differentiation).	The process of evaluating the net area under a curve between two specific points.
Representation	A family of functions (all antiderivatives + C).	A numerical value (a specific number).
Notation	∫ f(x) dx	∫ₐᵇ f(x) dx
Result	A function of x + an arbitrary constant C. E.g., ∫ 2x dx = x² + C	A pure number. E.g., ∫₁² 2x dx = 3
Limits/Bounds	No upper or lower limits of integration.	Has specific upper (b) and lower (a) limits.
Constant of Integration (C)	Must be included. It represents the family of all curves with the same derivative.	Not included. The constant cancels out when evaluating F(b) - F(a).
Geometric Meaning	Represents a family of parallel curves, each vertically shifted by C.	Represents the net signed area bounded by the curve y = f(x), the x-axis, and the lines x = a and x = b.
Objective	To find the general antiderivative.	To calculate a specific quantity like area, volume, displacement, etc.

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Definite and Indefinite Integration

1.0Introduction to Integration

Integration is the inverse process of differentiation.

If: $\frac{d y}{d x} = f (x)$ , then: $y = \int f (x) d x$

Integration is classified into two types:

Indefinite Integration → Represents a family of functions (with constant of integration).
Definite Integration → Represents a numerical value, usually area under a curve within limits.

2.0Indefinite Integration

Definition

The indefinite integral of a function f(x) is written as: $\int f (x) d x = F (x) + C$

where:

F′(x)=f(x)
C = constant of integration

Properties of Indefinite Integration

Linearity: $\int [a f (x) + b g (x)], d x = a \int f (x), d x + b \int g (x), d x$
Reversal of Differentiation: $\frac{d}{d x} (\int f (x), d x) = f (x)$
Constant Factor Rule: $\int a \cdot f (x), d x = a \int f (x), d x$

Standard Formulas

$\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C (n \neq = - 1)$
$\int \frac{1}{x} d x = ln ∣ x ∣ + C$
$\int e^{x} d x = e^{x} + C$
$\int a^{x} d x = \frac{a ^{x}}{l n a} + C$
$\int sin x d x = - cos x + C$
$\int cos x d x = sin x + C$
$\int sec^{2} x d x = tan x + C$
$\int csc^{2} x d x = - cot x + C$
$\int sec x tan x d x = sec x + C$
$\int csc x cot x d x = - csc x + C$

Rules of Integration

Linearity: $\int [a f (x) + b g (x)] d x = a \int f (x) d x + b \int g (x) d x$
Substitution Rule: If x=ϕ(t), then: $\int f (x) d x = \int f (ϕ (t)) ϕ^{'} (t) d t$
Integration by Parts: $\int u v d x = u \int v d x - \int (\frac{d u}{d x} \int v d x) d x$
Partial Fractions: Used when integrand is a rational function.

Methods of Indefinite Integration

Substitution Method

If ( f(x) ) is complex, substitute ( x ) with another variable ( t ) to simplify the integral.

Example:

$\int 2 x cos (x^{2}), d x$

Let $(t = x^{2} \Rightarrow d t = 2 x d x)$

$\int cos t, d t = sin t + C = sin (x^{2}) + C$

Integration by Parts

Used when integrating the product of two functions: [ $\int u, d v = uv - \int v, d u$ ] Choose ( u ) and ( dv ) wisely (using ILATE rule: Inverse, Log, Algebraic, Trig, Exponential).

Partial Fractions

Decompose rational functions into simpler fractions before integrating.

Example: $\int \frac{1}{x ^{2} - 1}, d x = \int (\frac{1}{2 ( x - 1 )} - \frac{1}{2 ( x + 1 )}) d x$

Solved Examples on Indefinite Integration

Example 1: $\int (3 x^{2} + 2 x + 1) d x = x^{3} + x^{2} + x + C$

Example 2: $\int \frac{1}{x ^{2} + 1} d x = tan^{- 1} x + C$

Example 3 (By Parts): $\int x e^{x} d x = x e^{x} - \int e^{x} d x = (x - 1) e^{x} + C$

Explore in Detail: Indefinite Integration

3.0Definite Integration

Definition of Definite Integration

The definite integral of f(x) between a and b is defined as: $\int_{a}^{b} f (x) d x = F (b) - F (a)$ , where F(x) is the antiderivative of f(x).

Properties of Definite Integrals

Additivity over Intervals: $\int_{a}^{b} f (x), d x + \int_{b}^{c} f (x), d x = \int_{a}^{c} f (x), d x$
Reversal of Limits: $\int_{a}^{b} f (x), d x = - \int_{b}^{a} f (x), d x$
Zero Interval: $\int_{a}^{a} f (x), d x = 0$
Linearity: $\int_{a}^{b} [a f (x) + b g (x)], d x = a \int_{a}^{b} f (x), d x + b \int_{a}^{b} g (x), d x$
Symmetry: For even and odd functions over ([-a, a]):

Even: ( $\int - a^{a} f (x), d x = 2 \int_{0}^{a} f (x), d x$ )
Odd: ( $\int_{- a}^{a} f (x), d x = 0$ )

Evaluation Techniques

Direct Substitution: Find the antiderivative, then substitute the limits.
Change of Variable: If substitution is used, adjust the limits accordingly.
Splitting the Integral: Break into simpler intervals if necessary.

Application of Definite Integrals

Area under Curves: The area between the curve ( y = f(x) ), the ( x )-axis, and vertical lines ( x = a ) and ( x = b ) is: $A = \int_{a}^{b} f (x), d x$
Area between Curves: For two curves ( y = f(x) ) and ( y = g(x) ), where ( $f (x) \geq g (x)$ ) in ( [a, b] ): $A = \int_{a}^{b} [f (x) - g (x)], d x$
Other Applications

Physics: Calculation of displacement, work done, etc.
Probability: Continuous probability distributions.
Average Value: $Average value of f (x) over [a, b] = \frac{1}{b - a} \int_{a}^{b} f (x), d x$

Solved Examples on Definite Integrals

Example 1: $\int_{0}^{1} x^{2} d x = [\frac{x ^{3}}{3}]_{0}^{1} = \frac{1}{3}$

Example 2 (Even Function): $\int_{- 2}^{2} x^{2} d x = 2 \int_{0}^{2} x^{2} d x = 2 [\frac{x ^{3}}{3}]_{0}^{2} = \frac{16}{3}$

Example 3 (Substitution): $\int_{0}^{π /2} sin x d x = [- cos x]_{0}^{π /2} = 1$

Example 4 (Property): $\int_{0}^{1} ln (\frac{1}{x}) d x = \int_{0}^{1} ln (\frac{1}{1 - x}) d x$

Learn in detail : Definite Integration

4.0Difference Between Definite and Indefinite Integration

Feature	Indefinite Integration	Definite Integration
Definition	The process of finding the antiderivative (the inverse of differentiation).	The process of evaluating the net area under a curve between two specific points.
Representation	A family of functions (all antiderivatives + C).	A numerical value (a specific number).
Notation	∫ f(x) dx	∫ₐᵇ f(x) dx
Result	A function of x + an arbitrary constant C. E.g., ∫ 2x dx = x² + C	A pure number. E.g., ∫₁² 2x dx = 3
Limits/Bounds	No upper or lower limits of integration.	Has specific upper (b) and lower (a) limits.
Constant of Integration (C)	Must be included. It represents the family of all curves with the same derivative.	Not included. The constant cancels out when evaluating F(b) - F(a).
Geometric Meaning	Represents a family of parallel curves, each vertically shifted by C.	Represents the net signed area bounded by the curve y = f(x), the x-axis, and the lines x = a and x = b.
Objective	To find the general antiderivative.	To calculate a specific quantity like area, volume, displacement, etc.

Frequently Asked Questions

What is the main difference between definite and indefinite integrals?

Why is +C added in indefinite integration?

Can definite integrals be negative?

Which methods are most important for JEE?

Frequently Asked Questions

What is the main difference between definite and indefinite integrals?

Why is +C added in indefinite integration?

Can definite integrals be negative?

Which methods are most important for JEE?

Join ALLEN!

Definite and Indefinite Integration

1.0Introduction to Integration

2.0Indefinite Integration

Definition

Properties of Indefinite Integration

Standard Formulas

Rules of Integration

Methods of Indefinite Integration

Solved Examples on Indefinite Integration

3.0Definite Integration

Definition of Definite Integration

Properties of Definite Integrals

Evaluation Techniques

Application of Definite Integrals

Solved Examples on Definite Integrals

4.0Difference Between Definite and Indefinite Integration

Table of Contents

Join ALLEN!

Definite and Indefinite Integration

1.0Introduction to Integration

2.0Indefinite Integration

Definition

Properties of Indefinite Integration

Standard Formulas

Rules of Integration

Methods of Indefinite Integration

Solved Examples on Indefinite Integration

3.0Definite Integration

Definition of Definite Integration

Properties of Definite Integrals

Evaluation Techniques

Application of Definite Integrals

Solved Examples on Definite Integrals

4.0Difference Between Definite and Indefinite Integration

Table of Contents