Evaluate the integrals in Exercises 37–44.
∫ sec²(θ) sin³(θ) dθ

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Rewrite the integral \( \int \sec^{2}(\theta) \sin^{3}(\theta) \, d\theta \) by expressing \( \sin^{3}(\theta) \) as \( \sin(\theta) \cdot \sin^{2}(\theta) \). This helps to use trigonometric identities to simplify the integrand.

Use the Pythagorean identity \( \sin^{2}(\theta) = 1 - \cos^{2}(\theta) \) to rewrite \( \sin^{3}(\theta) \) as \( \sin(\theta)(1 - \cos^{2}(\theta)) \). Substitute this back into the integral.

Rewrite the integral as \( \int \sec^{2}(\theta) \sin(\theta) (1 - \cos^{2}(\theta)) \, d\theta \). Now, consider a substitution to simplify the integral. Since \( \cos(\theta) \) and \( \sin(\theta) \, d\theta \) are related, let \( u = \cos(\theta) \).

Calculate \( du = -\sin(\theta) \, d\theta \), which implies \( -du = \sin(\theta) \, d\theta \). Substitute \( u \) and \( du \) into the integral, and also express \( \sec^{2}(\theta) \) in terms of \( u \) if possible, or rewrite \( \sec^{2}(\theta) \) as \( \frac{1}{\cos^{2}(\theta)} = \frac{1}{u^{2}} \).

Rewrite the integral entirely in terms of \( u \) and \( du \), then simplify the expression. After simplification, integrate with respect to \( u \). Finally, substitute back \( u = \cos(\theta) \) to express the answer in terms of \( \theta \).

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Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They help simplify integrals by rewriting expressions, such as converting powers of sine or cosine into more manageable forms. For example, using identities like sin²(θ) = 1 - cos²(θ) can facilitate integration.

Integration Techniques for Trigonometric Functions

Integrating trigonometric functions often requires techniques like substitution, rewriting powers, or using standard integral formulas. Recognizing when to apply substitution, such as setting u = cos(θ) or u = sin(θ), can simplify the integral. Familiarity with integrals of sec²(θ) and other trig functions is essential.

Substitution Method

The substitution method involves changing variables to simplify an integral. By choosing a substitution that relates to part of the integrand, the integral can be transformed into a basic form. For example, substituting u = cos(θ) when integrating expressions involving sin(θ) and cos(θ) can make the integral easier to solve.

Evaluate the integrals in Exercises 37–44.∫ sec²(θ) sin³(θ) dθ