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

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Recognize that the integral involves powers of sine and cosine with different arguments, so start by expressing \( \sin^3(\theta) \) in a form that is easier to integrate. Use the identity \( \sin^3(\theta) = \sin(\theta) \cdot \sin^2(\theta) \) and then rewrite \( \sin^2(\theta) \) using the Pythagorean identity \( \sin^2(\theta) = 1 - \cos^2(\theta) \).

Rewrite the integral as \( \int \sin(\theta) (1 - \cos^2(\theta)) \cos(2\theta) \, d\theta \). This separates the powers and prepares the integral for substitution or further simplification.

Use the double-angle identity for cosine: \( \cos(2\theta) = 2\cos^2(\theta) - 1 \). Substitute this into the integral to express everything in terms of \( \cos(\theta) \) and \( \sin(\theta) \).

Consider the substitution \( u = \cos(\theta) \), which implies \( du = -\sin(\theta) d\theta \). Rewrite the integral in terms of \( u \) and \( du \), replacing \( \sin(\theta) d\theta \) 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 \( u \).

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Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They simplify expressions and integrals by rewriting powers or products of sine and cosine into more manageable forms, such as using double-angle or power-reduction formulas.

Integration of Trigonometric Functions

Integrating trigonometric functions often requires rewriting the integrand using identities to express it in terms of basic sine and cosine functions. This process makes it easier to apply standard integral formulas and find antiderivatives.

Power-Reducing and Double-Angle Formulas

Power-reducing formulas convert powers of sine or cosine into expressions involving first powers of cosine or sine with multiple angles. Double-angle formulas express functions like cos(2θ) in terms of cos²θ and sin²θ, aiding in simplifying integrals involving products of trigonometric powers.

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