Checking Antiderivative Formulas

Right, or wrong? Say which for each formula and give a brief reason for each answer.

∫tanθ sec²θ dθ = sec³θ / 3 + C

검증된 단계별 안내

Identify the integral to be checked: \(\int \tan\theta \sec^{2}\theta \, d\theta\).

Recall the derivatives of common trigonometric functions: for example, \(\frac{d}{d\theta}(\sec\theta) = \sec\theta \tan\theta\) and \(\frac{d}{d\theta}(\sec^{3}\theta) = 3 \sec^{2}\theta \cdot \sec\theta \tan\theta = 3 \sec^{3}\theta \tan\theta\).

Compare the integrand \(\tan\theta \sec^{2}\theta\) with the derivative of \(\sec^{3}\theta\): since \(\frac{d}{d\theta}(\sec^{3}\theta) = 3 \sec^{3}\theta \tan\theta\), the integrand is \(\frac{1}{3}\) of this derivative.

Therefore, the integral \(\int \tan\theta \sec^{2}\theta \, d\theta\) should be \(\frac{1}{3} \sec^{3}\theta + C\), matching the given formula \(\frac{\sec^{3}\theta}{3} + C\).

Conclude that the formula is correct because the derivative of the proposed antiderivative matches the integrand.

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주요 개념

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Antiderivatives and Indefinite Integrals

An antiderivative of a function is another function whose derivative equals the original function. Indefinite integrals represent the family of all antiderivatives and include a constant of integration, C. Understanding how to find antiderivatives is essential for verifying integral formulas.

Integration by Substitution

Integration by substitution simplifies integrals by changing variables to make the integral easier to solve. It often involves identifying a part of the integrand as a derivative of another function. Recognizing substitution opportunities helps verify or find antiderivatives involving composite functions.

Derivatives of Trigonometric Functions

Knowing the derivatives of trigonometric functions like tan(θ) and sec(θ) is crucial for checking antiderivatives. For example, the derivative of sec(θ) is sec(θ)tan(θ), and the derivative of sec³(θ) involves the chain rule. This knowledge allows one to differentiate the proposed antiderivative to confirm correctness.

Checking Antiderivative FormulasRight, or wrong? Say which for each formula and give a brief reason for each answer.∫tanθ sec²θ dθ = sec³θ / 3 + C