The following list is intended to help you decide which integration technique to use for a given problem. The methods are presented in order of priority, i.e. from the first method to try to the last resort. 1. Simple substitution You should always check this first. 2. Partial fractions Is the integrand of the form P(x)/Q(x) where P(x) and Q(x) are polynomials? If so, use partial fractions. If not, go to the next method. 3. Powers of sin and cos Is the integrand of the form sin(x)^n cos(x)^m ? If so, use this method. If not, go to the next one. a. if one of n,m is odd, let u be the other function - the one raised to the even power. b. if both are odd, you can choose either sin or cos for u. c. if n and m are even, you have to use the half-angle formulas. 4. Does the integrand contain a term like sqrt(a^2-b^2x^2) ? If so, use the substitution x = a/b sin(theta). Then dx = a/b cos(theta) dtheta and a^2-b^2x^2 = a^2 cos(theta)^2. 5. If none of the first four methods are applicable, try integration by parts. Remember that your goal is to make the integral simpler and that you must choose dv to be something you can integrate.