Home | 18.013A | Chapter 19 | | Tools Glossary Index Up Previous Next

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

Up Previous Next