What are the basic rules of integration?

The basic rules include constant rule, power rule, sum rule, difference rule, substitution, and integration by parts.

How is integration useful in JEE exams?

Integration is used in solving problems of areas, volumes, probability, and calculus-based applications.

What is the difference between definite and indefinite integration?

Indefinite integral gives a family of functions with constant C. A definite integral gives a numerical value (area under curve).

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Integration Rules: Formulas, Properties, Methods & Solved Examples

1.0Introduction to Integration

Integration is one of the most important topics to learn in Calculus, and it is a good base for JEE Main and JEE Advanced problems. It is thought to be the opposite of differentiation.

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

y=∫f(x) dx+C

Here,

∫f(x)dx = integral of f(x) with respect to x
C = constant of integration

2.0Basics of Integration

Integration is used to find area under curves, volume of solids, and in solving differential equations.
Every function has infinitely many antiderivatives differing by a constant C.
Example: $\int x^{2} d x = \frac{x ^{3}}{3} + C .$

3.0Standard Integration Formulas

Some important formulas for JEE:

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

4.0Fundamental Rules of Integration

Rule 1: Constant Rule

$\int k d x = k x + C$

Where k is a constant.

Rule 2: Power Rule

$\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C (n \neq = - 1)$

Rule 3: Sum Rule

$\int [f (x) + g (x)] d x = \int f (x) d x + \int g (x) d x$

Rule 4: Difference Rule

$\int [f (x) - g (x)] d x = \int f (x) d x - \int g (x) d x$

Rule 5: Scalar Multiplication Rule

$\int k \cdot f (x) d x = k \int f (x) d x$

Rule 6: Integration of Zero Function

$\int 0 d x = C$

Rule 7: Integration of Exponential Functions

$\int e^{x} d x = e^{x} + C$

$\int a^{x} d x = \frac{a ^{x}}{l n a} + C$

Rule 8: Integration of Logarithmic Functions

$\int \frac{1}{x} d x = ln ∣ x ∣ + C$

Rule 9: Integration of Trigonometric Functions

$\int sin x d x = - cos x + C$

$\int cos x d x = sin x + C$

$\int sec^{2} x d x = tan x + C$

Rule 10: Substitution Rule

If x=g(t), then:

$\int f (g (t)) g^{'} (t) d t = \int f (x) d x$

Example:

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

$Let \overset{=}{˘} x^{2} ⟹ d u = 2 x d x$

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

Rule 11: Integration by Parts

$\int uv d x = u \int v d x - \int (\frac{d u}{d x} \int v d x) d x$

Useful for functions like x sin⁡x, xe^x, etc.

Rule 12: Integration of Rational Functions (Partial Fractions)

For rational functions:

$\int \frac{P ( x )}{Q ( x )} d x$

We decompose into partial fractions for easier integration.

5.0Properties of Definite Integrals

$\int_{a}^{a} f (x) d x = 0$
$\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x$
$\int_{a}^{b} [f (x) + g (x)] d x = \int_{a}^{b} f (x) d x + \int_{a}^{b} g (x) d x$
$If f (x) is even: \int_{- a}^{a} f (x) d x = 2 \int_{0}^{a} f (x) d x$

$\int_{- a}^{a} f (x) d x = 2 \int_{0}^{a} f (x) d x$

5. If f(x) is odd:

$\int_{- a}^{a} f (x) d x = 0$

6.0Solved Examples on Integration Rule

Example 1

Evaluate $\int (3 x^{2} + 2 x + 1) d x$

Solution:

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

Example 2

Evaluate $\int e^{2 x} d x$

Solution:

$\int e^{2 x} d x = \frac{1}{2} e^{2 x} + C$

Example 3

Evaluate $\int \frac{1}{x ^{2} + 1} d x$

Solution:

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

Example 4

Evaluate ∫xcos⁡x dx.

Solution (By Parts):
Let u=x,dv=cos ⁡xdx.

$\int x cos x d x = x sin x - \int 1 \cdot sin x d x$

$= x sin x + cos x + C$

Example 5

Evaluate $\int \frac{1}{( x - 1 ) ( x + 2 )} d x$

Solution (Partial Fractions):

$\frac{1}{( x - 1 ) ( x + 2 )} = \frac{A}{x - 1} + \frac{B}{x + 2}$

1 = A(x+2) + B(x-1)

Solving: $A = \frac{1}{3}, B = - \frac{1}{3}$

$\int \frac{1}{( x - 1 ) ( x + 2 )} d x = \frac{1}{3} ln ∣ x - 1∣ - \frac{1}{3} ln ∣ x + 2∣ + C$

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Integration Rules: Formulas, Properties, Methods & Solved Examples

1.0Introduction to Integration

Integration is one of the most important topics to learn in Calculus, and it is a good base for JEE Main and JEE Advanced problems. It is thought to be the opposite of differentiation.

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

y=∫f(x) dx+C

Here,

∫f(x)dx = integral of f(x) with respect to x
C = constant of integration

2.0Basics of Integration

Integration is used to find area under curves, volume of solids, and in solving differential equations.
Every function has infinitely many antiderivatives differing by a constant C.
Example: $\int x^{2} d x = \frac{x ^{3}}{3} + C .$

3.0Standard Integration Formulas

Some important formulas for JEE:

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

4.0Fundamental Rules of Integration

Rule 1: Constant Rule

$\int k d x = k x + C$

Where k is a constant.

Rule 2: Power Rule

$\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C (n \neq = - 1)$

Rule 3: Sum Rule

$\int [f (x) + g (x)] d x = \int f (x) d x + \int g (x) d x$

Rule 4: Difference Rule

$\int [f (x) - g (x)] d x = \int f (x) d x - \int g (x) d x$

Rule 5: Scalar Multiplication Rule

$\int k \cdot f (x) d x = k \int f (x) d x$

Rule 6: Integration of Zero Function

$\int 0 d x = C$

Rule 7: Integration of Exponential Functions

$\int e^{x} d x = e^{x} + C$

$\int a^{x} d x = \frac{a ^{x}}{l n a} + C$

Rule 8: Integration of Logarithmic Functions

$\int \frac{1}{x} d x = ln ∣ x ∣ + C$

Rule 9: Integration of Trigonometric Functions

$\int sin x d x = - cos x + C$

$\int cos x d x = sin x + C$

$\int sec^{2} x d x = tan x + C$

Rule 10: Substitution Rule

If x=g(t), then:

$\int f (g (t)) g^{'} (t) d t = \int f (x) d x$

Example:

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

$Let \overset{=}{˘} x^{2} ⟹ d u = 2 x d x$

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

Rule 11: Integration by Parts

$\int uv d x = u \int v d x - \int (\frac{d u}{d x} \int v d x) d x$

Useful for functions like x sin⁡x, xe^x, etc.

Rule 12: Integration of Rational Functions (Partial Fractions)

For rational functions:

$\int \frac{P ( x )}{Q ( x )} d x$

We decompose into partial fractions for easier integration.

5.0Properties of Definite Integrals

$\int_{a}^{a} f (x) d x = 0$
$\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x$
$\int_{a}^{b} [f (x) + g (x)] d x = \int_{a}^{b} f (x) d x + \int_{a}^{b} g (x) d x$
$If f (x) is even: \int_{- a}^{a} f (x) d x = 2 \int_{0}^{a} f (x) d x$

$\int_{- a}^{a} f (x) d x = 2 \int_{0}^{a} f (x) d x$

5. If f(x) is odd:

$\int_{- a}^{a} f (x) d x = 0$

6.0Solved Examples on Integration Rule

Example 1

Evaluate $\int (3 x^{2} + 2 x + 1) d x$

Solution:

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

Example 2

Evaluate $\int e^{2 x} d x$

Solution:

$\int e^{2 x} d x = \frac{1}{2} e^{2 x} + C$

Example 3

Evaluate $\int \frac{1}{x ^{2} + 1} d x$

Solution:

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

Example 4

Evaluate ∫xcos⁡x dx.

Solution (By Parts):
Let u=x,dv=cos ⁡xdx.

$\int x cos x d x = x sin x - \int 1 \cdot sin x d x$

$= x sin x + cos x + C$

Example 5

Evaluate $\int \frac{1}{( x - 1 ) ( x + 2 )} d x$

Solution (Partial Fractions):

$\frac{1}{( x - 1 ) ( x + 2 )} = \frac{A}{x - 1} + \frac{B}{x + 2}$

1 = A(x+2) + B(x-1)

Solving: $A = \frac{1}{3}, B = - \frac{1}{3}$

$\int \frac{1}{( x - 1 ) ( x + 2 )} d x = \frac{1}{3} ln ∣ x - 1∣ - \frac{1}{3} ln ∣ x + 2∣ + C$

Frequently Asked Questions

What are the basic rules of integration?

How is integration useful in JEE exams?

What is the difference between definite and indefinite integration?

Frequently Asked Questions

What are the basic rules of integration?

How is integration useful in JEE exams?

What is the difference between definite and indefinite integration?

Join ALLEN!

Integration Rules: Formulas, Properties, Methods & Solved Examples

1.0Introduction to Integration

2.0Basics of Integration

3.0Standard Integration Formulas

4.0Fundamental Rules of Integration

5.0Properties of Definite Integrals

6.0Solved Examples on Integration Rule

Example 1

Example 2

Example 3

Example 4

Example 5

Table of Contents

Join ALLEN!

Integration Rules: Formulas, Properties, Methods & Solved Examples

1.0Introduction to Integration

2.0Basics of Integration

3.0Standard Integration Formulas

4.0Fundamental Rules of Integration

5.0Properties of Definite Integrals

6.0Solved Examples on Integration Rule

Example 1

Example 2

Example 3

Example 4

Example 5

Table of Contents