Techniques of Integration

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

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

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.

Bill Farr <>
Last modified: Sat Sep 9 07:38:43 1995