Question

Exercises 59–64 require the use of various trigonometric identities before you evaluate the integrals.∫ sin³(θ) cos(2θ) dθ

Accepted Answer

Recognize that the integral involves powers of sine and cosine with different arguments, so start by expressing $$ \sin^3(	heta)  in a $$form that is easier to integrate. Use the identity $$ \sin^3(	heta) = \sin(	heta) \cdot \sin^2(	heta) $$ and then rewrite $$ \sin^2(	heta) $$ using the Pythagorean identity $$ \sin^2(	heta) = 1 - \cos^2(	heta) . $$Rewrite the integral as $$ \int \sin(	heta) (1 - \cos^2(	heta)) \cos(2	heta) \, d	heta . $$This separates the powers and prepares the integral for substitution or further simplification. Use the double-angle identity for cosine: $$ \cos(2	heta) = 2\cos^2(	heta) - 1 . $$Substitute this into the integral to express everything in terms of $$ \cos(	heta) $$ and $$ \sin(	heta) . $$Consider the substitution $$ u = \cos(	heta) $$, which implies $$ du = -\sin(	heta) d	heta . $$Rewrite the integral in terms of $$ u $$ and $$ du $$, replacing $$ \sin(	heta) d	heta $$ with $$ -du . $$After substitution, the integral becomes a polynomial in $$ u . $$Expand and simplify the polynomial expression, then integrate term-by-term with respect to $$ u .$$

Exercises 59–64 require the use of various trigonometric identities before you evaluate the integrals.
∫ sin³(θ) cos(2θ) dθ

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Key Concepts

Trigonometric Identities

Integration of Trigonometric Functions

Power-Reducing and Double-Angle Formulas