# Introduction

## Integrals class 12 of Class 12

### Definition

If f and F are functions of x such that F′(x) = f(x) then the function F is called a primitive or an indefinite integral of f with respect to x and is expressed as ∫ f(x) dx = F(x) + c where c is a constant of integration.

f(x) then possesses infinitely many antiderivatives, all of them being contained in the expression F(x) + c where c is a constant.

From the geometric point of view, an indefinite integral is a collection (family) of curves, each of which is obtained by translating one of the curves parallel to itself upwards or downwards.

### Expression Substitution

√a^{2} – x^{2} x = a sinθ or a conθ

√a^{2} – x^{2} x = a tanθ or a cotθ

√a^{2} – x^{2} x = a tanθ or a cotθ

√a^{2} – x^{2} x = a secθ or a cosecθ

√a+x/a-x or √a-x/a+x x = a conθ or a cos2θ

√2ax – x^{2} x = a (1 – cosθ)

###
**Critical Points**

The points on the curve y = f(x) at which dy/dx = 0 or dy/dx does not exist are called critical points.

- Introduction
- Properties of Indefinite Integration
- Some Results In Integration
- Methods of Integration
- Use of Eulers Theorem
- Cancellation of Integrals
- Integration of Rational Functions
- Integration of Trigonometric Functions
- Integration of Basic Irrational Functions
- Definite Integral
- Properties of Definite Integrals
- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6
- Exercise 7