76–83. Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals.
76. ∫ [cosθ / (sin³θ - 4sinθ)] dθ

Verified step by step guidance

1. Identify a suitable substitution to simplify the integral. Notice that the denominator involves powers of \( \sin \theta \), so let \( u = \sin \theta \).
2. Compute the differential \( du \) in terms of \( d\theta \). Since \( u = \sin \theta \), then \( du = \cos \theta \, d\theta \).
3. Rewrite the integral in terms of \( u \) and \( du \). The numerator \( \cos \theta \, d\theta \) becomes \( du \), and the denominator \( \sin^3 \theta - 4 \sin \theta \) becomes \( u^3 - 4u \). So the integral becomes \( \int \frac{du}{u^3 - 4u} \).
4. Factor the denominator \( u^3 - 4u \) to prepare for partial fraction decomposition. Factor out \( u \) to get \( u(u^2 - 4) \), and recognize \( u^2 - 4 \) as a difference of squares: \( (u - 2)(u + 2) \).
5. Set up the partial fraction decomposition for \( \frac{1}{u(u - 2)(u + 2)} \) and solve for the constants. Then integrate each term separately with respect to \( u \).

Key Concepts

Change of Variables (Substitution)

This technique involves replacing a complicated expression with a simpler variable to make integration easier. For example, substituting u = sinθ can transform the integral into a rational function in terms of u, simplifying the integrand before further methods are applied.

Partial Fraction Decomposition

Partial fractions break down a complex rational expression into simpler fractions that are easier to integrate. After substitution, the integrand often becomes a rational function where this method helps separate terms for straightforward integration.

Trigonometric Identities and Simplification

Using identities like sin²θ + cos²θ = 1 or factoring expressions involving sine and cosine can simplify the integrand. Recognizing these patterns is essential before substitution or partial fractions to reduce the integral to a manageable form.