Calculus I · Unit 3A · lesson

Trigonometric Integrals

Concept

Learning objectives

Use parity and identities to integrate powers and products of sine, cosine, tangent, and secant.

Trigonometric Integrals

Explanation

Parity determines the useful identity

Integrals involving powers of sine, cosine, secant, or tangent are solved by exposing a derivative-compatible factor. For products of sine and cosine powers, an odd power often lets us save one factor and convert the rest using $\sin^2x+\cos^2x=1$ . When both powers are even, half-angle identities usually reduce the exponents.

The same pattern governs tangent and secant. A secant-squared factor pairs with the derivative of tangent, while a secant-tangent factor pairs with the derivative of secant. The identities are not random decorations; they reorganize the integrand into a substitution pattern. Before calculating, classify the powers and state which derivative pair you are trying to create.

For $\sin^m x\cos^n x$ :

• if one exponent is odd, save one factor of that function and convert the rest using $\sin^2x+\cos^2x=1$ ; • if both exponents are even, use power-reduction identities.

For powers of tangent and secant, preserve a $\sec^2x$ factor for $u=\tan x$ or a $\sec x\tan x$ factor for $u=\sec x$ when possible.

Worked example

Odd sine power

\int\sin^3x\cos^2x\,dx =\int(1-\cos^2x)\cos^2x\sin x\,dx.

Let $u=\cos x$ , $du=-\sin x\,dx$ :

-\int(u^2-u^4)\,du=-\frac{u^3}{3}+\frac{u^5}{5}+C.

Thus

-\frac{\cos^3x}{3}+\frac{\cos^5x}{5}+C.

Interactive checku3a-trig-int-01

Evaluate $\int\sin^2x\cos x\,dx$ .

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Let $u=\sin x$ .

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After the explanation

Use the section idea

Reading lens

Choose an integration method from the integrand's structure, then verify the result by differentiation.

Mental model

Substitution reverses a chain rule, parts reverses a product rule, and algebraic or trigonometric rewrites expose a recognizable derivative pattern.

Decision

Simplify first; look for an inner derivative; then consider parts, identities, trigonometric substitution, or partial fractions in a deliberate order.

Common trap

Choosing a method by surface appearance, transforming only part of the differential, or accepting a more complicated integral than the one you started with.

Check yourself

Can you name the derivative rule being reversed and differentiate the final answer back to the integrand?

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