First, start by simplifying the integrand as much as possible (using simple algebraic manipulations or basic trigonometric identities). Not necessarily a simpler form but more a form that we know how to integrate. Go over the reviews of basic integration techniques and common functions: The basics , Common functions.
Next, look for obvious substitution (a function whose derivative also occurs), one that will get you an integral that is easy to do. Here’s a review of simple substitution: Substitution
If you can’t solve the integral using simplification or substitution, it’s time to step up. Try to classify the integrand into one of the following: product of trig powers, rational functions, radicals, or a product of a polynomial and a transcendental function.
Trig functions (product of powers of trig functions: Trig functions, Advanced trig functions
Rational functions ( use partial fractions if the degree of the numerator is less than the degree of the denominator, otherwise use long division), here’s how it’s done: Partial fractions , Long division
Integration by parts (product of a polynomial and a transcendental function: Integration by parts
Radicals (use trig substitution if the integral contains sqrt(a^2+x^2) or sqrt(x^2-a^2), for (ax+b)^1/n try simple substitution ): Trig substitution , Advanced trig substitution
If none of the above techniques work, you should take some more aggressive measures; advanced algebraic manipulations, trig identities, integration by parts with no product (assume 1 as a multiplier). In some cases you need to use multiple techniques. Here are some examples of advanced integration: Advanced integration
The best strategy is to assume easy until easy doesn’t work, always try the simplest techniques first, and remember there is more than one way to solve an integral.