Integration by Parts

In this tutorial we shall develop a technique that will help us to evaluate a wide variety of integrals that do not fit any of the basic integration formulas. This technique of integration that is quite useful is integration by parts. It depends upon the formula for the derivative of a product.

\frac{d}{{dx}}\left[  {f\left( x \right)\int {g\left( x \right)dx} } \right] = f'\left( x \right)\int  {g\left( x \right)dx + f\left( x \right)\frac{d}{{dx}}\left[ {\int {g\left( x  \right)dx} } \right]}

\frac{d}{{dx}}\left[  {f\left( x \right)\int {g\left( x \right)dx} } \right] = f'\left( x \right)\int  {g\left( x \right)dx + f\left( x \right)g\left( x \right)}

Integrating both sides, we have

\begin{gathered} f\left( x \right)\int {g\left( x \right)dx =  \int {\left[ {f'\left( x \right)\int {g\left( x \right)dx} } \right]dx + \int  {f\left( x \right)g\left( x \right)dx} } } \\ \Rightarrow \int {f\left( x \right)g\left( x  \right)dx}  = f\left( x \right)\int  {g\left( x \right)dx - \int {\left[ {f'\left( x \right)\int {g\left( x  \right)dx} } \right]dx} } \,\,\,{\text{ -   -  - }}\left( {\text{i}} \right) \\ \end{gathered}

Formula (i) is called the formula for integration by parts. With the help of this formula we integrate the product of two functions. The basic idea to use this formula is that we shall consider the two functions as first and second functions.

The function whose integration can easily be found is considered as the second function while the other is considered as the first function. In the formula (i), f\left( x \right) and g\left( x \right) are considered as the first and second functions respectively. Sometimes both functions are easily integrable, in this case we may chose any of them as first function provided that the integrand does not becomes complicated.