# Concept of Anti Derivatives or Integration

The inverse process of derivatives is 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 is$f\left( {\text{x}} \right)$, that is,
$\frac{{\text{d}}}{{{\text{dx}}}}\left[ {\phi \left( {\text{x}} \right)} \right] = f\left( {\text{x}} \right)$

Thus if

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

Then

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

Note:

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