Integral of Derivative over Function

The 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, the integral sign $\int {}$ and $\frac{d}{{dx}}$ on the right side will cancel each other out, 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 of integration

, we have

where $A$ is the constant of integration.