Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [cos(θ) / (sin²(θ) + sin(θ) − 6)] dθ

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Start by examining the integral: \(\int \frac{\cos(\theta)}{\sin^{2}(\theta) + \sin(\theta) - 6} \, d\theta\). Notice that the denominator is a quadratic expression in terms of \(\sin(\theta)\).

Make the substitution \(u = \sin(\theta)\), which implies that \(du = \cos(\theta) \, d\theta\). This substitution will simplify the integral because \(\cos(\theta) \, d\theta\) can be replaced by \(du\).

Rewrite the integral in terms of \(u\): it becomes \(\int \frac{1}{u^{2} + u - 6} \, du\). Now the integral is a rational function in \(u\).

Factor the quadratic in the denominator: \(u^{2} + u - 6 = (u + 3)(u - 2)\). This allows you to use partial fraction decomposition to break the integral into simpler fractions.

Set up the partial fraction decomposition: \(\frac{1}{(u + 3)(u - 2)} = \frac{A}{u + 3} + \frac{B}{u - 2}\). Solve for constants \(A\) and \(B\), then integrate each term separately with respect to \(u\).

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Integration by Substitution

Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable, which transforms the integral into a simpler form. This technique is especially useful when the integral contains composite functions.

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Trigonometric Identities and Functions

Understanding trigonometric functions like sine and cosine and their properties is essential. Recognizing how to manipulate expressions involving sin(θ) and cos(θ) helps in simplifying the integrand or choosing an appropriate substitution. Familiarity with identities can also aid in rewriting the integral.

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Partial Fraction Decomposition

Partial fraction decomposition breaks down complex rational expressions into simpler fractions that are easier to integrate. When the denominator is a polynomial, factoring it allows the integral to be expressed as a sum of simpler terms. This method is often combined with substitution in trigonometric integrals.

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