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

\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 , so the integral sign \int {} and \frac{d}{{dx}} on the right side will cancel each other, 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 for 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.