Finding Indefinite Integrals
Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ sec θ/3 tan θ/3 dθ

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Recognize the integral to solve: \(\int \sec\left(\frac{\theta}{3}\right) \tan\left(\frac{\theta}{3}\right) \, d\theta\).

Recall the derivative identity: \(\frac{d}{dx} \sec x = \sec x \tan x\). This suggests that the integral of \(\sec x \tan x\) with respect to \(x\) is \(\sec x + C\).

Use a substitution to handle the argument \(\frac{\theta}{3}\). Let \(u = \frac{\theta}{3}\), so that \(d\theta = 3 \, du\).

Rewrite the integral in terms of \(u\): \(\int \sec u \tan u \cdot 3 \, du = 3 \int \sec u \tan u \, du\).

Integrate with respect to \(u\) using the known formula: \(\int \sec u \tan u \, du = \sec u + C\). Then substitute back \(u = \frac{\theta}{3}\) to express the answer in terms of \(\theta\).

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Indefinite Integrals

An indefinite integral represents the most general antiderivative of a function, expressed with a constant of integration (C). It reverses differentiation and provides a family of functions whose derivative is the integrand. Understanding indefinite integrals is essential for solving problems without specified limits.

Integration Techniques for Trigonometric Functions

Integrating trigonometric functions often requires recognizing standard forms or using substitution. For example, integrals involving secant and tangent functions can be simplified by recalling derivatives like d/dx(sec x) = sec x tan x. Familiarity with these relationships helps in guessing and verifying antiderivatives.

Verification by Differentiation

After finding an antiderivative, differentiating it should return the original integrand. This step confirms the correctness of the solution. Checking answers by differentiation is a crucial practice to ensure no mistakes were made during integration.

Finding Indefinite Integrals Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation. ∫ sec θ/3 tan θ/3 dθ