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.

∫ cos³ 𝓍/2 d𝓍

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Rewrite the integral to make it easier to handle. Use the identity for powers of cosine: express \(\cos^3\left(\frac{x}{2}\right)\) as \(\cos\left(\frac{x}{2}\right) \cdot \cos^2\left(\frac{x}{2}\right)\).

Recall the Pythagorean identity: \(\cos^2\theta = 1 - \sin^2\theta\). Substitute this into the integral to get \(\int \cos\left(\frac{x}{2}\right) \left(1 - \sin^2\left(\frac{x}{2}\right)\right) dx\).

Use substitution to simplify the integral. Let \(u = \sin\left(\frac{x}{2}\right)\), then find \(du\) in terms of \(dx\). Since \(u = \sin\left(\frac{x}{2}\right)\), differentiate to get \(du = \frac{1}{2} \cos\left(\frac{x}{2}\right) dx\), or equivalently, \(\cos\left(\frac{x}{2}\right) dx = 2 du\).

Rewrite the integral in terms of \(u\): it becomes \(\int (1 - u^2) \cdot 2 du = 2 \int (1 - u^2) du\). This integral is straightforward to evaluate.

Integrate term-by-term: \(2 \int (1 - u^2) du = 2 \left( u - \frac{u^3}{3} \right) + C\). Finally, substitute back \(u = \sin\left(\frac{x}{2}\right)\) to express the antiderivative in terms of \(x\).

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주요 개념

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Indefinite Integral

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.

Integration Techniques for Powers of Trigonometric Functions

Integrating powers of trigonometric functions often requires rewriting the integrand using identities or substitution. For example, expressing cos³(x/2) as cos(x/2)·cos²(x/2) and then using the identity cos²θ = 1 - sin²θ simplifies the integral.

Verification by Differentiation

After finding an antiderivative, differentiating it should return the original integrand. This step confirms the correctness of the indefinite integral and helps identify any errors in the integration process.

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. ∫ cos³ 𝓍/2 d𝓍