Integration by Parts

The formula

When a function is a product of two simpler ones, integration by parts breaks it down:

\int u\,dv = uv - \int v\,du

The trick is picking which factor is $u$ (something that gets simpler when differentiated) and which is $dv$ (something easy to integrate).

Find $\int x\cos x\,dx$ .

Solution: Let $u = x$ and $dv = \cos x\,dx$ , so $du = dx$ and $v = \sin x$ :

\int x\cos x\,dx = x\sin x - \int \sin x\,dx = x\sin x - (-\cos x) + C

= x\sin x + \cos x + C

Integration by Parts

When a function is a product of two simpler ones, integration by parts breaks it down:

\int u\,dv = uv - \int v\,du

The trick is picking which factor is $u$ (something that gets simpler when differentiated) and which is $dv$ (something easy to integrate).

Find $\int x\cos x\,dx$ .

Solution: Let $u = x$ and $dv = \cos x\,dx$ , so $du = dx$ and $v = \sin x$ :

\int x\cos x\,dx = x\sin x - \int \sin x\,dx = x\sin x - (-\cos x) + C

= x\sin x + \cos x + C