Integration by parts

Integration by parts (IBP) is an integration technique. It is best used in cases with some polynomial term and an exponential or trigonometric function, or multiple trigonometric functions. This works for both indefinite and definite integrals but you have to be mindful of applying bounds.

$$\int udv=uv-\int vdu$$

The basic principle is we “exchange” a difficult integral for one that is easier to solve. Our choice for $u$ should be the function that’s easiest to differentiate, and our choice for $dv$ the easiest to integrate.

Basic examples

For $\int x\sin (x)dx$:

$$u=x;dv=\sin (x)dx$$ $$du=dx;v=-\cos (x)$$

So:

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

For $\int \ln (x)dx$:

$$u=\ln (x);dv=(1)dx$$ $$du=\frac{1}{x}dx;v=x$$

So:

$$\int \ln (x)dx=x\ln (x)-x+C$$

Cyclic integration

We would loop forever if we did IBP for certain integrands, specifically combinations of exponential and trigonometric functions. This doesn’t mean we can’t solve them.

Try $\cos x\sin x$ as the integrand.

$$I=\int \cos x\sin xdx$$ $$I=\int \cos x\sin xdx={\sin }^{2}x-\int \cos x\sin xdx$$ $$I={\sin }^{2}x-I$$ $$2I={\sin }^{2}x$$ $$I=\frac{{\sin }^{2}x}{2}$$

See also

• Integrands involving the product of an exponential and trigonometric term can be solved an alternate way using complex analysis. See this page.