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.

∫7sin(θ/3) dθ

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Identify the integral to solve: \(\int 7 \sin\left(\frac{\theta}{3}\right) \, d\theta\).

Recognize that the integral involves a sine function with a linear argument \(\frac{\theta}{3}\). Use substitution to simplify the integral. Let \(u = \frac{\theta}{3}\), so that \(\theta = 3u\).

Calculate the differential \(d\theta\) in terms of \(du\): since \(\theta = 3u\), then \(d\theta = 3 \, du\).

Rewrite the integral in terms of \(u\): \(\int 7 \sin(u) \, d\theta = \int 7 \sin(u) \cdot 3 \, du = \int 21 \sin(u) \, du\).

Integrate \(21 \sin(u)\) with respect to \(u\): recall that \(\int \sin(u) \, du = -\cos(u) + C\), so the integral becomes \(-21 \cos(u) + C\). Finally, substitute back \(u = \frac{\theta}{3}\) to express the answer in terms of \(\theta\).

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주요 개념

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Indefinite Integral

An indefinite integral represents the most general antiderivative of a function, including a constant of integration. It reverses differentiation and is expressed without limits, showing all possible functions whose derivative matches the integrand.

Integration of Trigonometric Functions

Integrating trigonometric functions like sine involves using known antiderivatives, such as ∫sin(ax) dx = -cos(ax)/a + C. Recognizing the inner function and applying the chain rule in reverse is essential for correct integration.

Substitution Method

The substitution method simplifies integration by changing variables to handle composite functions. For example, setting u = θ/3 transforms the integral into a simpler form, making it easier to integrate and then revert to the original variable.

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.∫7sin(θ/3) dθ