How to choose an identity for a trigonometric integral
The parity of sine, cosine, secant, and tangent powers tells you what to save and what to convert.
A practical comparison that turns a vague choice into a repeatable test.
Reviewed July 11, 2026For sine and cosine, save one factor when a power is odd; use power-reduction identities when both are even. For secant and tangent, save sec² for tangent substitution or sec·tan for secant substitution.
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.
Evaluate ∫ sin³x cos²x dx.
Save one sine and convert the even remainder.
The saved factor becomes the differential.
Integrate a polynomial in u.
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.
Three takeaways
- Parity drives the first choice.
- Save a factor that becomes du.
- Use identities to convert the remaining expression to one trig function.