Integrals of trigonometric functions appear throughout calculus because sine, cosine, and tangent model waves, rotations, alternating current, sound, and many periodic motions. These integrals often look difficult at first because products and powers of trig functions do not always match basic antiderivative rules. The main skill is recognizing patterns and choosing an identity or substitution that turns the expression into something familiar.
A strategy map helps you decide whether to save a factor, use a power-reduction identity, convert with secant and tangent, or simplify first.
For powers of sine and cosine, odd powers usually suggest saving one factor and using sin^2 x + cos^2 x = 1, while even powers usually suggest half-angle identities. For powers of tangent and secant, the choice often depends on whether a sec^2 x or sec x tan x factor can be saved for substitution. Products such as sin(mx)cos(nx) can often be simplified using product-to-sum identities before integrating.
These methods are not separate tricks, but organized ways to create a substitution pair or reduce powers until a basic antiderivative applies.
Key Facts
- Basic antiderivatives: ∫sin x dx = -cos x + C and ∫cos x dx = sin x + C.
- Pythagorean identities: sin^2 x + cos^2 x = 1, 1 + tan^2 x = sec^2 x, and 1 + cot^2 x = csc^2 x.
- If sin^m x cos^n x has an odd sine power, save one sin x and use sin^2 x = 1 - cos^2 x with u = cos x.
- If sin^m x cos^n x has an odd cosine power, save one cos x and use cos^2 x = 1 - sin^2 x with u = sin x.
- Power-reduction identities: sin^2 x = (1 - cos 2x)/2 and cos^2 x = (1 + cos 2x)/2.
- Useful tangent-secant strategy: if sec^n x tan^m x has an even secant power, save sec^2 x and use tan^2 x = sec^2 x - 1 with u = tan x.
Vocabulary
- Antiderivative
- An antiderivative of a function f(x) is a function F(x) whose derivative is f(x).
- Trigonometric identity
- A trigonometric identity is an equation involving trig functions that is true for all allowed input values.
- Power-reduction identity
- A power-reduction identity rewrites a power such as sin^2 x or cos^2 x using a lower power and a double angle.
- u-substitution
- u-substitution is an integration method that reverses the chain rule by replacing part of an integrand with a new variable.
- Product-to-sum identity
- A product-to-sum identity rewrites a product of trig functions as a sum or difference of trig functions.
Common Mistakes to Avoid
- Treating ∫sin^2 x dx as (1/3)sin^3 x is wrong because integration does not undo powers that way unless the derivative of the inside is present.
- Forgetting to save one sine or cosine factor in an odd-power integral is wrong because the saved factor usually supplies du for substitution.
- Using sin^2 x = 1 - cos x instead of sin^2 x = 1 - cos^2 x is wrong because the missing square changes the identity and the entire integrand.
- Dropping the constant C in an indefinite integral is wrong because every antiderivative represents a family of functions that differ by a constant.
Practice Questions
- 1 Evaluate ∫sin^3 x cos^2 x dx.
- 2 Evaluate ∫sec^4 x tan^3 x dx.
- 3 Decide which strategy should be used first to evaluate ∫sin^4 x cos^2 x dx, and explain why an odd-power substitution is not the best first step.