Use identities to expose an inner derivative

Try

Differentiate sin⁴x/4 in radians. The result is sin³x cos x, revealing a direct substitution. Now remove the cosine factor and consider sin²x. The same substitution no longer removes every x-dependent term simply. Predict why a different trigonometric identity may be needed.

Predict

For a product of powers of sine and cosine, predict what happens if one power is odd. You can reserve one factor for a differential and convert the remaining even power using sin²x+cos²x=1. If both powers are even, half-angle identities provide another route.

Experiment

Transform ∫sin²x cos³x dx systematically.

Step	Expression
Reserve one cosine factor	sin²x cos²x · cos x dx
Replace the even cosine power	sin²x(1−sin²x) cos x dx
Set u=sin x	(u²−u⁴)du
Integrate and return	sin³x/3−sin⁵x/5+C

Differentiate the result and factor out sin²x cos x. The remaining factor is 1−sin²x=cos²x, restoring the original integrand. This verification also confirms that no negative sign or inner derivative has been lost.

Understand

For an odd sine power, reserve sin x dx and substitute u=cos x, remembering du=−sin x dx. For an odd cosine power, reserve cos x dx and substitute u=sin x. The remaining even power can be converted by the Pythagorean identity. These are useful patterns, not compulsory procedures; a different valid simplification may be shorter.

For sin²x, use sin²x=(1−cos 2x)/2. Its primitive is x/2−sin(2x)/4+C. The denominator 4 includes both the identity’s factor 1/2 and the factor 1/2 from integrating cos 2x. Differentiation gives 1/2−cos(2x)/2, which is the original square. For cos²x, the sign before cos 2x is positive. Integrating either square over [0,π] yields π/2, consistent with their equal average over that interval.

Tangent and secant products use a related pair of derivatives: (tan x)′=sec²x and (sec x)′=sec x tan x. In ∫tan x sec²x dx, take u=tan x. In ∫sec³x tan x dx, reserve sec x tan x dx and integrate the remaining sec²x as u². The result is sec³x/3+C. All these formulas apply on intervals where cosine is nonzero; crossing a pole is a separate improper-integral issue.

Even secant powers can often be reduced by reserving sec²x and using sec²x=1+tan²x. Odd tangent powers may permit a secant substitution after using tan²x=sec²x−1. Some cases, including sec³x alone, require integration by parts rather than these simplest patterns. Method choice should reduce complexity and preserve domains, not just resemble a memorized example.

Angles are in radians throughout. Standard trigonometric derivative rules encode this convention. A function written in degree-number input has extra conversion factors and cannot be integrated with these formulas unchanged. Keep π symbolic in exact answers rather than converting it to a rounded decimal before evaluating bounds.

Apply

For each practice problem, state the identity or reserved differential that enables the method. Convert definite bounds when using a new variable. For a symbolic primitive, differentiate it and show the identity that recovers the integrand. Explain any interval exclusions caused by tangent or secant poles.

Continue

The next lesson uses trigonometric substitutions to simplify algebraic radicals. The new challenge is choosing an angle branch so square roots have the correct nonnegative sign.