Trigonometric substitution replaces x with a trig function to get rid of a square root. Use x = a sin θ for √(a² − x²), x = a tan θ for √(a² + x²) and x = a sec θ for √(x² − a²). A Pythagorean identity then turns the root into a single trig term.
Trigonometric integrals are integrals of powers and products of sin x, cos x, tan x and sec x. If sine or cosine has an odd power, save one factor, change the rest with sin²x + cos²x = 1, and substitute. If both powers are even, use the half-angle identities to lower the powers first.
Integration by parts is a method for integrating a product of two functions. It uses the formula ∫u dv = uv − ∫v du. Choose u to be the part that gets simpler when you differentiate it, and dv to be the rest.