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Home | 18.013A | Chapter 19 |
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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 and we can write as , then we can write as , and an anti-derivative of is then the difference between any anti-derivative of and one of .
But an anti-derivative of is given by ; so we can use the product rule here to reduce the problem of finding an anti-derivative of to finding an anti-derivative of , for any and .
This tool is useful for finding anti-derivatives of products of the form if you know an anti-derivative for and an anti-derivative for as well.
We can set and in the identity above, and write , which by this procedure is , where we have used the identity . This has as an anti-derivative and therefore an anti-derivative of .
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 , that is, we seek the antiderivative of with respect to .
If we set
and
, we can deduce that
and
is a possible antiderivative of
.
Integrating by parts tells us then
which gives, after integrating
Exercises:
Try integrating the following integrands with respect to by using this technique:
19.1.
19.2.
19.3.
19.4. (Hint: integrate by parts twice and solve the resulting equation.)
19.5.
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