__BASIC CONCEPT__

Let F(x) be a differentiable function of x such that

Then F(x) is called the integral of f(x). Symbolically, it is written as ∫ f(x) dx = F(x).

f(x) , the function to be integrated , is called the integrand.

F(x) is also called the anti-derivative (or primitive function) of f(x).

__Constant of Integration:__

As the differential coefficient of a constant is zero, we have

Therefore, ∫f(x) dx = F(x) + c.

This constant c is called the constant of integration and can take any real value

__Properties of Indefinite Integration:__

(i)

( Here ‘ a ‘ is a constant)

(ii)

(iii) If ∫ f(u)du = F(u) + c , then ∫f(ax + b) dx =

__Integration as the Inverse Process of Differentiation :__

**Evaluate :**

(i) ∫tan^{2}x dx

(ii) ∫e^{x}(cosx − sinx)dx

(iii) ∫3x^{1/2}(1 + x^{3/2})dx

(iv) ∫sin3x cos2x dx

Solutions: (i) ∫tan^{2}x dx = ∫(sec^{2}x − 1)dx = tan x − x + c.

(ii) ∫e^{x}(cosx − sinx)dx.

Here e^{x}(cosx − sinx) is the derivative of e^{x} cosx.

=> I = e^{x}cosx + c.

(iii) ∫3x^{1/2}(1 + x^{3/2})dx.

Here 3x^{1/2}(1 + x^{3/2}) is the derivative of (1 + x^{3/2})^{2}

=> I = (1 + x^{3/2})^{2} + c.

(iv) ∫sin3x cos2x dx

When solving such problems it is expedient to use the following trigonometric identities :

sin(mx) cos(nx) = (1/2) [sin(m − n)x + sin(m + n)x]

sin(mx) sin(nx) = (1/2) [cos(m − n)x − cos(m + n)x]

cos(mx) cos(nx) = (1/2) [cos(m − n)x + cos(m + n)x]

Here sin 3x cos 2x = (1/2) [sin (5x) + sin x]

=> I =

**Exercise 1:**

Integrate the following functions

(i)

(ii)

(iii)

(iv)

__FORMULAE :__

**Basic formulae**