How do I work out what integration method I should use to solve an integral?

A-Level Mathematics is about a wide range of topics, among which Integrating is key!

While differentiation is a technique, whereby a set of rules can be used to approach problems, integration is an art, for which no one can provide you with a practical recipe.

In this sense, experience is fundamental to solving integrals, this is why practice is really the only way to 'learn' how to integrate.

The following line of reasoning should be of guidance in how to approach an integral and looking for its evaluation.

Simplify the integrand, if possible. This step is very important in the integration process. Many integrals can be taken from impossible or very difficult to very easy with a little simplification or manipulation. Don’t forget basic trig and algebraic identities as these can often be used to simplify the integral.

See if a “simple” substitution will work. Look to see if a simple substitution can be used

Identify the type of integral. Note that any integral may fall into more than one of these types. Because of this fact it’s usually best to go all the way through the list and identify all possible types since one may be easier than the other and it’s entirely possible that the easier type is listed lower in the list.
  Is the integrand a rational expression (i.e is the integrand a polynomial divided by a polynomial)? If so, then partial fractions may work on the integral.Is the integrand a polynomial times a trig function, exponential, or logarithm? If so, then integration by parts may work.Is the integrand a product of sines and cosines, secant and tangents, or cosecants and cotangents? If so, then the topics from the second section may work.
Likewise, don’t forget that some quotients involving these functions can also be done using these techniques.Does the integrand involve , , or ? If so, then a trig substitution might work nicely.Does the integrand have roots other than those listed above in it? If so, then the substitution might work.Does the integrand have a quadratic in it? If so, then completing the square on the quadratic might put it into a form that we can deal with.

Can we relate the integral to an integral we already know how to do? Do we need to use multiple techniques? In this step we need to ask ourselves if it is possible that we’ll need to use multiple techniques.

Try again. If everything that you’ve tried to this point doesn’t work then go back through the process and try again. This time try a technique that that you didn’t use the first time around.