Educator

Integration of Trigonometric Functions

Main formula:

Hints and tips:

• Remember to check which of the powers is odd. Then let u be the other one. (E.g. if you have cos⁵ x, then use u = sin x.)

• If both powers are even, use the half-angle formulas.

• If you have an even power of secant, use u = tan x. If you have an odd power of tangent, use u = sec x.

• If you have an odd power of secant and an even power of tangent, then follow this process:

1. Convert tangents to secants, two at a time, using tan² +1 = sec² . You’ll end up with odd powers of secant.

2. Use integration by parts twice (u = secⁿ⁻² x, dv = sec² x dx) to cut down from ∫ secⁿ x dx to ∫ secⁿ⁻² x dx, so you eventually get back to ∫ sec x dx. Then remember the formula: