Evaluate the integrals in Exercises 39–54.
∫ sin(θ) dθ / (cos²θ + cos θ - 2)

Verified step by step guidance

1. Start by examining the integral: \(\int \frac{\sin(\theta)}{\cos^{2}(\theta) + \cos(\theta) - 2} \, d\theta\). Notice that the denominator is a quadratic expression in terms of \(\cos(\theta)\).
2. Let \(u = \cos(\theta)\). Then, compute \(du = -\sin(\theta) \, d\theta\), which implies \(-du = \sin(\theta) \, d\theta\). This substitution will help simplify the integral.
3. Rewrite the integral in terms of \(u\): replace \(\sin(\theta) \, d\theta\) with \(-du\), and the denominator becomes \(u^{2} + u - 2\). So the integral becomes \(\int \frac{-du}{u^{2} + u - 2}\).
4. Factor the quadratic in the denominator: \(u^{2} + u - 2 = (u + 2)(u - 1)\). This allows you to use partial fraction decomposition to express \(\frac{1}{(u + 2)(u - 1)}\) as a sum of simpler fractions.
5. Set up the partial fractions: \(\frac{1}{(u + 2)(u - 1)} = \frac{A}{u + 2} + \frac{B}{u - 1}\). Solve for constants \(A\) and \(B\), then integrate each term separately with respect to \(u\).

Key Concepts

Integration of Trigonometric Functions

This involves techniques to integrate functions containing sine, cosine, and other trigonometric expressions. Recognizing patterns and using identities can simplify the integral, making it easier to solve.

Trigonometric Identities and Factorization

Using identities like factoring quadratic expressions in terms of cosine helps simplify the denominator. For example, factoring cos²θ + cosθ - 2 into (cosθ + 2)(cosθ - 1) can make the integral more manageable.

Substitution Method in Integration

Substitution involves changing variables to simplify the integral. Here, letting u = cosθ transforms the integral into a rational function in u, which is easier to integrate.