INDEFINITE INTEGRAL

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) ∫tan2x dx

(ii) ∫ex(cosx − sinx)dx

(iii) ∫3x1/2(1 + x3/2)dx

(iv) ∫sin3x cos2x dx

Solutions: (i) ∫tan2x dx = ∫(sec2x − 1)dx = tan x − x + c.

(ii) ∫ex(cosx − sinx)dx.

Here ex(cosx − sinx) is the derivative of ex cosx.

=> I = excosx + c.

(iii) ∫3x1/2(1 + x3/2)dx.

Here 3x1/2(1 + x3/2) is the derivative of (1 + x3/2)2

=> I = (1 + x3/2)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

Standard Formulae:

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