How to select the correct integration method?

Question

I often find my self lost and doing the same thing when solving math problems. My professor seems to know exactly what will work and what will not in an integration problem, how do I do that?

For example, consider this: $e^{-x^2}$

when I see this i would look at the integration methods I know, pick the ones I can apply, and try them all (Very time consuming). So I would try u substitution first

But there are $3$ different possible choices for u! $u = x^2$ or $u = x$ or $u = e^{-x^2}$

I would try these, then move on to the by parts method, trig substitution, partial fractions, and so on. Do you see where my problem is? It's making integration so frustrating to do!

sure, it is sometimes obvious what Not to use (so partial fractions here is a no no), but still I end up with so many possible choices, and every choice branches off into even more choices!

Sometimes I try to skip some methods, only to find that they were the solution!

How do I predict what will work and what will not?

Answer 1

With enough practice in u-sub, you can "try" a substitution without writing anything down. Figure out the derivative of your substitution, and see if you can force it into the integral, while converting all $x$ to $u$. If you're having trouble doing this process quickly, become more efficient with differentiation.

Then, if this doesn't work, move on to integration by parts. Again, with practice, you should be able to check this without writing anything.

Of course, you only do the above steps if you have no idea how to solve an integral. Typically, you will know what method you should use based on the form of the integral. I would never try a trig sub on $e^{x^2}$, mostly because I know that the integral is impossible to solve.

Answer 2

The strategy that I suggest is as follows.

1. First check is the anti-derivative in a table of anti-derivatives.

2. If not try a manipulation. Expanding products, rewriting surds, trig identities, partial fractions, etc.

3. If things are still not in the tables you need to try a $u$-substitution. From reading your question my guess is that you need help on $u$-substitution which is the Chain Rule ran backwards:

$$\int f'(g(x))\cdot kg'(x)\,dx=k\cdot f(g(x))+C.$$

You need to be able to pick $u=g(x)$. There are three ways:

• see the function --- multiple-of-derivative pattern and let $u=$ 'function'.
• the $u$ is 'inside' another function (the first function in the composition)...
• chance LIATE (better for integration by parts).

Note that the substitution will never be $u=x$... this will just change all $x$s to $u$s (this advice does not hold for integration by parts).

4. If a $u$-substitution doesn't work you can try integration by parts.

Once you can master these four techniques you can add a 5., 6., etc. but you can't really go very far until you can do $u$-substitutions no fuss.

Practise is key.

Note: This isn't guaranteed to work. For the anti-derivative in the question:

$$\int e^{x^2}\,dx,$$

there is no anti-derivative in terms of elementary functions so none of these methods will help.