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.