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
then,
Where C is the constant of integration.
In Class 12 Maths, integration is a core chapter under Calculus. It includes:
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.
2. Definite Integrals:
These represent a fixed numerical value over a given interval [a, b]:
Integral calculus deals with finding integrals and solving problems involving:
Here are some basic integral calculus formulas frequently used:
This theorem connects differentiation and integration:
It states that integration and differentiation are inverse operations.
It also helps evaluate definite integrals easily:
Where F(x) is the antiderivative of f(x).
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:
(Session 2025 - 26)