Checking Antiderivative Formulas

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

∫xsinx dx = -x cos x + C

검증된 단계별 안내

Recall the integration by parts formula: \(\int u \, dv = uv - \int v \, du\).

Identify parts for integration by parts: let \(u = x\) (so \(du = dx\)) and \(dv = \sin x \, dx\) (so \(v = -\cos x\)).

Apply the formula: \(\int x \sin x \, dx = -x \cos x - \int (-\cos x) \, dx = -x \cos x + \int \cos x \, dx\).

Integrate \(\int \cos x \, dx\) to get \(\sin x\), so the full antiderivative is \(-x \cos x + \sin x + C\).

Compare this with the given formula \(\int x \sin x \, dx = -x \cos x + C\); since the \(\sin x\) term is missing, the given formula is incorrect.

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주요 개념

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Antiderivative (Indefinite Integral)

An antiderivative of a function f(x) is another function F(x) whose derivative is f(x). The indefinite integral symbol ∫ represents the family of all antiderivatives, including a constant of integration C, since differentiation of a constant is zero.

Integration by Parts

Integration by parts is a technique used to integrate products of functions. It is based on the product rule for differentiation and follows the formula ∫u dv = uv - ∫v du, where u and dv are parts of the original integrand chosen to simplify the integral.

Verification of Antiderivatives

To verify if a given formula is an antiderivative, differentiate the proposed function and check if the result matches the original integrand. This method confirms correctness by reversing the integration process.

Checking Antiderivative FormulasRight, or wrong? Say which for each formula and give a brief reason for each answer.∫xsinx dx = -x cos x + C