Definite and Indefinite Integration

1.0Introduction to Integration

Integration is the inverse process of differentiation.

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

Integration is classified into two types:

1. Indefinite Integration → Represents a family of functions (with constant of integration).
2. 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: $\int f(x)dx=F(x)+C$

where:

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

Properties of Indefinite Integration

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

Standard Formulas

• $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C(n\ne -1)$
• $\int \frac{1}{x}dx=\ln |x|+C$
• $\int e^{x}dx=e^x+C$
• $\int a^{x}dx=\frac{a^x}{\ln a}+C$
• $\int \sin xdx=-\cos x+C$
• $\int \cos xdx=\sin x+C$
• $\int {\sec }^{2}xdx=\tan x+C$
• $\int {\csc }^{2}xdx=-\cot x+C$
• $\int \sec x\tan xdx=\sec x+C$
• $\int \csc x\cot xdx=-\csc x+C$

Rules of Integration

1. Linearity: $\int [af(x)+bg(x)]dx=a\int f(x)dx+b\int g(x)dx$
2. Substitution Rule: If x=ϕ(t), then: $\int f(x)dx=\int f(\varphi (t)){\varphi }^{\prime }(t)dt$
3. Integration by Parts:$\int uvdx=u\int vdx-\int \left(\frac{du}{dx}\int vdx\right)dx$
4. 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:

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

Let $(t=x^2\Rightarrow dt=2xdx)$

$\int \cos t,dt=\sin t+C=\sin (x^2)+C$

Integration by Parts

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

Partial Fractions

Decompose rational functions into simpler fractions before integrating.

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

Solved Examples on Indefinite Integration

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

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

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

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: ${\int }_a^{b}f(x)dx=F(b)-F(a)$, where F(x) is the antiderivative of f(x).

Properties of Definite Integrals

1. Additivity over Intervals: ${\int }_a^{b}f(x),dx+{\int }_b^{c}f(x),dx={\int }_a^{c}f(x),dx$
2. Reversal of Limits: ${\int }_a^{b}f(x),dx=-{\int }_b^{a}f(x),dx$
3. Zero Interval: ${\int }_a^{a}f(x),dx=0$
4. Linearity: ${\int }_a^b[af(x)+bg(x)],dx=a{\int }_a^{b}f(x),dx+b{\int }_a^{b}g(x),dx$
5. Symmetry: For even and odd functions over ([-a, a]):

• Even: ($\int {-a}^{a}f(x),dx=2{\int }_0^{a}f(x),dx$)
• Odd: (${\int }_{-a}^{a}f(x),dx=0$)

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

1. Area under Curves: The area between the curve ( y = f(x) ), the ( x )-axis, and vertical lines ( x = a ) and ( x = b ) is: $A={\int }_a^{b}f(x),dx$
2. Area between Curves: For two curves ( y = f(x) ) and ( y = g(x) ), where ($f(x)\ge g(x)$) in ( [a, b] ): $A={\int }_a^b[f(x)-g(x)],dx$
3. Other Applications

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

Solved Examples on Definite Integrals

Example 1: ${\int }_0^1{x}^{2}dx={\left[\frac{x^3}{3}\right]}_0^1=\frac{1}{3}$

Example 2 (Even Function): ${\int }_{-2}^2{x}^{2}dx=2{\int }_0^2{x}^{2}dx=2{\left[\frac{x^3}{3}\right]}_0^2=\frac{16}{3}$

Example 3 (Substitution): ${\int }_0^{\pi /2}\sin xdx={\left[-\cos x\right]}_0^{\pi /2}=1$

Example 4 (Property): ${\int }_0^1\ln \left(\frac{1}{x}\right)dx={\int }_0^1\ln \left(\frac{1}{1-x}\right)dx$

Learn in detail : Definite Integration

4.0Difference Between Definite and Indefinite Integration