Finding Indefinite Integrals

In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫csc θ/(csc θ − sin θ) dθ

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Start by rewriting the integrand to express everything in terms of sine and cosine functions. Recall that \(\csc \theta = \frac{1}{\sin \theta}\). So rewrite the integral as \(\int \frac{\frac{1}{\sin \theta}}{\frac{1}{\sin \theta} - \sin \theta} \, d\theta\).

Simplify the denominator by finding a common denominator inside it: \(\frac{1}{\sin \theta} - \sin \theta = \frac{1 - \sin^2 \theta}{\sin \theta}\). Use the Pythagorean identity \(1 - \sin^2 \theta = \cos^2 \theta\) to rewrite the denominator as \(\frac{\cos^2 \theta}{\sin \theta}\).

Now the integrand becomes \(\frac{\frac{1}{\sin \theta}}{\frac{\cos^2 \theta}{\sin \theta}} = \frac{1}{\sin \theta} \times \frac{\sin \theta}{\cos^2 \theta} = \frac{1}{\cos^2 \theta} = \sec^2 \theta\).

Recognize that the integral simplifies to \(\int \sec^2 \theta \, d\theta\). Recall the antiderivative of \(\sec^2 \theta\) is \(\tan \theta + C\).

Write the most general antiderivative as \(\tan \theta + C\), where \(C\) is the constant of integration. To verify, differentiate \(\tan \theta + C\) and confirm you get back the original integrand.

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Indefinite Integral and Antiderivative

An indefinite integral represents the most general antiderivative of a function, including an arbitrary constant of integration. It reverses differentiation, finding a function whose derivative matches the integrand. Understanding this concept is essential for solving integrals without specified limits.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values in their domains. Simplifying integrals often requires rewriting expressions using identities like csc θ = 1/sin θ or combining terms to facilitate integration. Mastery of these identities helps transform complex integrands into manageable forms.

Verification by Differentiation

After finding an antiderivative, verifying the solution by differentiating it ensures correctness. Differentiation should return the original integrand, confirming the integral was computed properly. This step is crucial for validating answers in indefinite integral problems.

Finding Indefinite IntegralsIn Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.∫csc θ/(csc θ − sin θ) dθ