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) , 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
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)
Integration as the Inverse Process of Differentiation :
= tanx – x + c
Here ex(cosx − sinx) is the derivative of ex cosx.
=> I = excosx + c.
Here 3x1/2(1 + x3/2) is the derivative of (1 + x3/2)2
=> I = (1 + x3/2)2 + c.
NOTE : When solving such problems it is expedient to use the following trigonometric identities :
Integrate the following functions
Basic formulae :