What is Integration in Maths? Types, Formulas & Class 12 Guide

What is Integration in Maths?

Integration is one of the most essential concepts in calculus, especially in Class 12 Maths and higher studies. It is essential in resolving problems related to areas, volumes, and accumulation of quantities. Integration is often referred to as the reverse process of differentiation.

In simple terms, integration is a mathematical technique to find the area under a curve or to accumulate quantities. It combines small parts to form a whole. It is commonly applied in physics, engineering, economics, and various real-life applications.

Mathematically, integration is the inverse operation of differentiation.

If $\frac{d}{d x} F (x) = f (x)$

then,

$\int f (x) d x = F (x) + C$

Where C is the constant of integration.

1.0Integration in Class 12 Maths

In Class 12 Maths, integration is a core chapter under Calculus. It includes:

Indefinite Integration: General integration without limits.
Definite Integration: Integration over a specific interval to find exact values.
Applications of Integrals: Finding areas bounded by curves.

2.0Types of Integrals

There are mainly two types of integrals in mathematics:

1. Indefinite Integrals:

Indefinite integrals represent a family of functions, and they include a constant of integration, C.

$\int f (x) d x = F (x) + C$

2. Definite Integrals:

These represent a fixed numerical value over a given interval [a, b]:

$\int_{a}^{b} f (x) d x$

3.0Integration Calculus (Integral Calculus)

Integral calculus deals with finding integrals and solving problems involving:

Areas under curves
Volumes of solids of revolution
Accumulation of quantities
Motion and displacement problems

4.0Integral Calculus Formulas

Here are some basic integral calculus formulas frequently used:

Function	Integral
$\int x^{n} d x$	$\frac{x ^{n + 1}}{n + 1} + C (for n \neq = - 1)$
$\int \frac{1}{x} d x$	ln +C
$\int e^{x} d x$	$e^{x} + C$
$\int sin x d x$	$- cos x + C$
$\int cos x d x$	$sin x + C$

5.0Fundamental Theorem of Integral Calculus

This theorem connects differentiation and integration:

$\frac{d}{d x} (\int_{a}^{x} f (t) d t) = f (x)$

It states that integration and differentiation are inverse operations.

It also helps evaluate definite integrals easily:

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

Where F(x) is the antiderivative of f(x).

6.0Differential and Integral Calculus

Differential calculus focuses on finding the rate of change and slopes of curves using derivatives.
Integral calculus deals with the accumulation of quantities and area under curves through integrals.

Together, they form the two major branches of calculus:

Differential Calculus: Study of change.
Integral Calculus: Study of accumulation.