]> 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