Trigonometric Integrals
Trigonometric integrals involve products and powers of sin and cos. The strategy depends on whether the powers are odd or even.
Strategies for ∫sinᵐx cosⁿx dx
Odd Power Example
Medium
Evaluate ∫sin³x cos²x dx
1
Save one sin
= ∫sin²x cos²x · sin x dx
2
Convert sin²x
= ∫(1 - cos²x) cos²x · sin x dx
3
Substitute u = cos x
du = -sin x dx
= -∫(1 - u²)u² du = -∫(u² - u⁴) du
4
Integrate
= -(u³/3 - u⁵/5) + C = -cos³x/3 + cos⁵x/5 + C
Answer:
-cos³x/3 + cos⁵x/5 + C
Key Takeaways
Odd power: save one, convert rest with Pythagorean identity
Even powers: use half-angle formulas
sin²x + cos²x = 1 is your best friend