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
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