]> 19.5 Integration by Parts

## 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

$d ( f g ) d x = f d g d x + d f d x g$

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

But an anti-derivative of $( f g ) '$ is given by $f g$ ; so we can use the product rule here to reduce the problem of finding an anti-derivative of $f g '$ 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 $( B x ) ' − B$ , where we have used the identity $x ' = 1$ . This has $B x − C$ as an anti-derivative and $B x − C$ therefore an anti-derivative of $A x$ .

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 $x ln ⁡ x d x$ , that is, we seek the antiderivative of $x ln ⁡ x$ with respect to $x$ .

If we set $u = ln ⁡ x$ and $d v = x d x$ , we can deduce that $d u = d x x$ and $v = x 2 2$ is a possible antiderivative of $x$ .
Integrating by parts tells us then $u d v = ( u v ) ' − v d u$ which gives, after integrating

$∫ x ( ln ⁡ x ) d x = x 2 ln ⁡ x 2 − ∫ x 2 2 1 x d x = x 2 ln ⁡ x 2 − x 2 4 + C$

Exercises:

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

19.1. $x 4 ( ln ⁡ x )$

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. $x ( sin ⁡ x ) exp ⁡ x$