How to choose an identity for a trigonometric integral

Sine and cosine powers

An odd sine power lets you save one sin x and convert the remaining even power with sin²x = 1 − cos²x. Then u = cos x matches the saved differential.

If both powers are even, neither derivative is available directly, so half-angle identities reduce the powers.

Secant and tangent powers

An even secant power suggests saving sec²x and converting the rest with sec²x = 1 + tan²x. An odd tangent power often pairs with a saved sec x tan x after converting tangent squares.

Exceptional canonical integrals, especially sec³x, need their own integration-by-parts argument.

Identities should create a derivative

Do not expand trig expressions at random. The goal is to expose a factor that becomes du while expressing the rest in the matching variable.

Worked example

Common mistakes

• Using a half-angle identity when an odd power already provides a substitution.
• Saving a factor but failing to convert every remaining trig function.
• Assuming all secant powers follow the same pattern as tangent powers.

Keep these ideas

• Parity drives the first choice.
• Save a factor that becomes du.
• Use identities to convert the remaining expression to one trig function.