# Integral of Derivative over Function

Integration of derivative over function of $x$ is another important formula of integration.

The integration of derivative over function of $x$ is of the form

Now consider

Integrating both sides of equation (i) with respect to $x$, we have

Since integration and differentiation are reverse processes to each other , so the integral sign $\int {}$and $\frac{d}{{dx}}$ on the right side will cancel each other, i.e.

Example: Evaluate the integral $\int {\frac{{ax + \frac{1}{2}b}}{{a{x^2} + bx + c}}\,dx}$ with respect to $x$

We have integral

Using the formula for integration

, we have

Where $A$ is the constant of integration.

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