Calculus : Integration By Parts

Integration By Parts

Recall that we derived the formula for integration by substitution by using the the Chain Rule and integrating it using the fundamental theorem of calculus. You might reasonably ask "What happens if we use one of the other rules of differentiation?"

This would be an exceedingly good question, and it even has an answer. If we take the product rule:

and integrate both sides, we get:

This isn't particularly useful until we re-arrange and get:

If you let u=f(x) and v=g(x) , then we can abuse notation in the standard way and say that du=f'(x) and dv=g'(x). Then the above is re-written as:

which may be easier to remember.

How does this help us?

If we can split the integrand up into two bits:

something we can differentiate, ie. f(x),
and something we can integrate, ie. g'(x),

then we can apply the formula. The hope is that the new integral we get, ie. ∫gf' dx, will be easier to compute than the original integral.

Sometimes you will find that you need to apply the whole process again to the new integral, but if you do, be careful that you don't end up back where you started.

Unfortunately, choosing the correct way to split the integrand can be tricky to get right, and the only real way to get good at it is to practice. Initially you will make some missteps, but eventually you will sort out how to pick the right functions because you will be able to think ahead and guess what the final integral will look like.

Good ways to choose (although these are not written in stone) include:

polynomials make good choices for f, because they tend to get simpler as you differentiate them.
log, arctan and arcsin are good choices for f if there are polynomials around, since you may get the chance to cancel.
exponential functions, sin and cos make good choices for g', because they tend not to get more complicated when you integrate.
since integration is harder than differentiation, try and make g' as complex as you can while still making sure you can integrate it.
in general you should aim to be making the integrand that you get out "simpler" than the one you started with.
if at first you don't succeed, try a different way of splitting the integrand.

Definite Integrals

Integration by parts works without much additional difficulty when doing definite integrals. The limits of the second integral are unchanged, and what you get is the following:

Tricks of the Trade

There are a handful of special tricks that employ integration by parts to solve otherwise difficult integration problems. Each of these is illustrated by a typical example which explains the technique.

Multiplying by 1

Sometimes you will be confronted by an integrand that you can't see how to split, but which you can see how to differentiate. The classic example of this is:

The trick here is to think of this as:

and let f(x)=log x, and g'(x)=1.

Getting the Same Integral After Integration by Parts

Sometimes you may find that after doing integration by parts (possibly multiple times), you find yourself back at the integral you started with. For example:

can be differentiated by parts to get:

and again to get:

The trick here is to let

so that the expression becomes:

and we solve for I, to get:

This is perhaps the simplest possible case, but you can get more complex expressions involving your integral. The same trick will work, however. Be vary careful with this trick, since if you use it incorrectly you will end up with everything cancelling, which means that you chose the wrong u and dv in one of the integration by parts steps.

Reduction Formulas

Sometimes you will find that after you integrate by parts you have an almost identical integral, but one which differs from the original by some parameter. For example:

Notice how the 5 has become a 4. In these cases it is often easiest to find a general rule, which is called a reduction formula , to help you evaluate the integral. In this example, the general rule is:

and by using this general rule repeatedly, you get:

Notice that the integral gets reduced down to ∫e^x dx, which is an integral we can evaluate without needing integration by parts.

Sometimes you may need to combine this technique with the previous trick to get your reduction formula.