Concept of Anti Derivatives or Integration

The inverse process of derivatives called anti–derivatives or integration.
“A function f\left(  {\text{x}} \right)being given and it is required to find a Second function\phi \left( {\text{x}} \right) whose derivative with respect to x, isf\left(  {\text{x}} \right), that is,

\frac{{\text{d}}}{{{\text{dx}}}}\left[ {\phi \left(  {\text{x}} \right)} \right] = f\left( {\text{x}} \right)

Thus if

\frac{{\text{d}}}{{{\text{dx}}}}\left[ {\phi \left(  {\text{x}} \right)} \right] = f\left( {\text{x}} \right)


\int {f\left( {\text{x}} \right){\text{dx}} = \phi \left(  {\text{x}} \right) + {\text{c}}}

The function \phi  \left( {\text{x}} \right) + {\text{c}}, then, is called the anti-derivate or indefinite integral of f\left( {\text{x}}  \right).

Symbol of Integration:
The function f\left(  {\text{x}} \right)is called integrand and the symbol \int {} is called the integral sign.
Thus, \int {f\left(  {\text{x}} \right){\text{dx}}} means that f\left( {\text{x}} \right)is to be integrated with respect to (w.r.t) ‘x’ and is read as “Integral of f\left( {\text{x}} \right)”. Where dx indicates the variable with respect to which f\left(  {\text{x}} \right)is to be integrated.

  • In \phi \left( {\text{x}} \right) + c, where ‘c’ is called as the constant of integration.
  • \frac{{\text{d}}}{{{\text{dx}}}} and \int   \cdots  {\text{ dx}}are inverse operations of each other.