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 $\frac{dy}{dx}=f(x)$,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: $\int x^{2}dx=\frac{x^3}{3}+C.$

3.0Standard Integration Formulas

Some important formulas for JEE:

1. $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C(n\ne -1)$
2. $\int \frac{1}{x}dx=\ln |x|+C$
3. $\int e^{x}dx=e^x+C$
4. $\int a^{x}dx=\frac{a^x}{\ln a}+C$
5. $\int \sin xdx=-\cos x+C$
6. $\int \cos xdx=\sin x+C$
7. $\int {\sec }^{2}xdx=\tan x+C$
8. $\int {\csc }^{2}xdx=-\cot x+C$
9. $\int \sec x\tan xdx=\sec x+C$
10. $\int \csc x\cot xdx=-\csc x+C$

4.0Fundamental Rules of Integration

Rule 1: Constant Rule

$\int kdx=kx+C$

Where k is a constant.

Rule 2: Power Rule

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

Rule 3: Sum Rule

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

Rule 4: Difference Rule

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

Rule 5: Scalar Multiplication Rule

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

Rule 6: Integration of Zero Function

$\int 0dx=C$

Rule 7: Integration of Exponential Functions

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

$\int a^{x}dx=\frac{a^x}{\ln a}+C$

Rule 8: Integration of Logarithmic Functions

$\int \frac{1}{x}dx=\ln |x|+C$

Rule 9: Integration of Trigonometric Functions

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

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

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

Rule 10: Substitution Rule

If x=g(t), then:

$\int f(g(t))g^{\prime }(t)dt=\int f(x)dx$

Example:

$\int 2x\cos (x^2)dx$

$\text{Let }\breve{=}{x}^2⟹du=2xdx$

$\int 2x\cos (x^2)dx=\int \cos udu=\sin u+C=\sin (x^2)+C$

Rule 11: Integration by Parts

$\int uvdx=u\int vdx-\int \left(\frac{du}{dx}\int vdx\right)dx$

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

Rule 12: Integration of Rational Functions (Partial Fractions)

For rational functions:

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

We decompose into partial fractions for easier integration.

5.0Properties of Definite Integrals

1. ${\int }_a^{a}f(x)dx=0$
2. ${\int }_a^{b}f(x)dx=-{\int }_b^{a}f(x)dx$
3. ${\int }_a^b[f(x)+g(x)]dx={\int }_a^{b}f(x)dx+{\int }_a^{b}g(x)dx$
4. $\text{If }f(x)\text{ is even:}{\int }_{-a}^{a}f(x)dx=2{\int }_0^{a}f(x)dx$

${\int }_{-a}^{a}f(x)dx=2{\int }_0^{a}f(x)dx$

5. If f(x) is odd:

${\int }_{-a}^{a}f(x)dx=0$

6.0Solved Examples on Integration Rule

Example 1

Evaluate $\int (3x^2+2x+1)dx$

Solution:

$\int (3x^2+2x+1)dx=x^3+x^2+x+C$

Example 2

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

Solution:

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

Example 3

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

Solution:

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

Example 4

Evaluate ∫xcos⁡x dx.

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

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

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

Example 5

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

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

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