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. The technique of integration that is quite useful here 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 here is that we shall consider the two functions as the first and second functions.

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