19.5 Integration by Parts

The second useful tool is the backward version of the product rule. The product rule, as we have noted often, tells us

This means that if we seek an anti-derivative of h(x) and we can write h as fg', then we can write fg' as (fg)' - f 'g, and an anti-derivative of fg' is then the difference between any anti-derivative of (fg)' and one of f 'g.

But an anti-derivative of (fg)' is given by fg; so we can use the product rule here to reduce the problem of finding an anti-derivative of fg' to finding an anti-derivative of f 'g, for any f and g.

This tool is useful for finding anti-derivatives of products of the form A(x)x if you know an anti-derivative B(x) for A(x) and an anti-derivative C(x) for B(x) as well.

We can set f = x and g = B in the identity above, and write A(x)x = B'(x)x, which by this procedure is (Bx)' - B, where we have used the identity x' = 1. This has Bx - C as an anti-derivative and Bx - C is therefore an anti-derivative of Ax.

The procedure is called integration by parts. It is useful for finding anti-derivatives of products of exponentials and powers or of trigonometric functions and powers or of logarithms and powers, among other things.

For example, suppose we want to integrate x ln x dx, that is, we seek the antiderivative of x ln x with respect to x.

If we set u = ln x and dv = x dx, we can deduce that is a possible antiderivative of x.
Integrating by parts tells us then udv = (uv)' - vdu which gives, after integrating

Exercises:

Try integrating the following integrands with respect to x by using this technique:

19.1. x⁴(ln x)

19.2. x sin x

19.3. x exp x

19.4. (sin x) exp x (Hint: integrate by parts twice and solve the resulting equation.)

19.5. x (sin x) exp x