Integration by parts | Mathematico

Integration by parts allows us to integrate a product of two functions.

Recall that the product rule allows us to differentiate products of functions. Suppose

h (x) = f (x) g (x)

then

h^{'} (x) = f^{'} (x) g (x) + f (x) g^{'} (x)

But, because integration is the opposite of differentiation, this means

\begin{aligned} \int h^{'} (x) d x & = \int f^{'} (x) g (x) + f (x) g^{'} (x) d x \\ h (x) & = \int f^{'} (x) g (x) d x + \int f (x) g^{'} (x) d x \\ f (x) g (x) & = \int f^{'} (x) g (x) d x + \int f (x) g^{'} (x) d x \end{aligned}

If we rearrange this, we finally get the integration by parts formula:

\int f^{'} (x) g (x) d x = f (x) g (x) - \int f (x) g^{'} (x) d x

Let's see how this formula works with an example.

Use integration by parts to find