# 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
$\int {\frac{{f’\left( x \right)}}{{f\left( x \right)}}dx = } \ln f\left( x \right) + c$

Now consider
$\begin{gathered} \frac{d}{{dx}}\left[ {\ln f\left( x \right) + c} \right] = \frac{d}{{dx}}\ln f\left( x \right) + \frac{d}{{dx}}\left( c \right),\,\,\,\,f\left( x \right) > 0 \\ \Rightarrow \frac{d}{{dx}}\left[ {\ln f\left( x \right) + c} \right] = \frac{1}{{f\left( x \right)}}\frac{d}{{dx}}f\left( x \right) + 0 \\ \Rightarrow \frac{d}{{dx}}\left[ {\ln f\left( x \right) + c} \right] = \frac{1}{{f\left( x \right)}}f’\left( x \right) \\ \Rightarrow \frac{{f’\left( x \right)}}{{f\left( x \right)}} = \frac{d}{{dx}}\left[ {\ln f\left( x \right) + c} \right] \\ \Rightarrow \frac{{f’\left( x \right)}}{{f\left( x \right)}}\,dx = d\left[ {\ln f\left( x \right) + c} \right]\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right) \\ \end{gathered}$

Integrating both sides of equation (i) with respect to $x$, we have
$\int {\frac{{f’\left( x \right)}}{{f\left( x \right)}}\,dx} = \int {d\left[ {\ln f\left( x \right) + c} \right]}$

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.
$\int {\frac{{f’\left( x \right)}}{{f\left( x \right)}}\,dx} = \ln f\left( x \right) + c$

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

We have integral $I = \int {\frac{{ax + \frac{1}{2}b}}{{a{x^2} + bx + c}}\,dx}$
$\int {\frac{{ax + \frac{1}{2}b}}{{a{x^2} + bx + c}}\,dx} = \frac{1}{2}\int {\frac{{2ax + b}}{{a{x^2} + bx + c}}\,dx}$

Using the formula of integration $\int {\frac{{f’\left( x \right)}}{{f\left( x \right)}}\,dx} = \ln f\left( x \right) + c$, we have
$\int {\frac{{ax + \frac{1}{2}b}}{{a{x^2} + bx + c}}\,dx} = \frac{1}{2}\ln \left( {a{x^2} + bx + c} \right) + A$

where $A$ is the constant of integration.