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.