Checking Antiderivative Formulas

Right, or wrong? Give a brief reason why.

∫−15(x + 3)² / (x − 2)⁴ dx = ((x + 3)/(x − 2))³ + C

검증된 단계별 안내

Identify the given integral: \(\int -15 \frac{(x + 3)^2}{(x - 2)^4} \, dx\) and the proposed antiderivative: \(\left( \frac{x + 3}{x - 2} \right)^3 + C\).

To verify if the proposed antiderivative is correct, differentiate \(\left( \frac{x + 3}{x - 2} \right)^3\) using the chain rule.

Let \(u = \frac{x + 3}{x - 2}\). Then the derivative of \(u^3\) with respect to \(x\) is \(3u^2 \cdot \frac{du}{dx}\).

Find \(\frac{du}{dx}\) by applying the quotient rule: \(\frac{d}{dx} \left( \frac{x + 3}{x - 2} \right) = \frac{(1)(x - 2) - (x + 3)(1)}{(x - 2)^2} = \frac{x - 2 - x - 3}{(x - 2)^2} = \frac{-5}{(x - 2)^2}\).

Combine these results: \(\frac{d}{dx} \left( \frac{x + 3}{x - 2} \right)^3 = 3 \left( \frac{x + 3}{x - 2} \right)^2 \cdot \left( \frac{-5}{(x - 2)^2} \right) = -15 \frac{(x + 3)^2}{(x - 2)^4}\), which matches the integrand, confirming the antiderivative is correct.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Antiderivative and Indefinite Integration

An antiderivative of a function is another function whose derivative equals the original function. Indefinite integration finds all antiderivatives and includes a constant of integration (C) because differentiation loses constant terms. Verifying an antiderivative involves differentiating the proposed solution and checking if it matches the original integrand.

Chain Rule and Power Rule in Differentiation

The chain rule is used to differentiate composite functions, applying the derivative of the outer function multiplied by the derivative of the inner function. The power rule states that d/dx[x^n] = n*x^(n-1). Understanding these rules is essential to verify if the given antiderivative is correct by differentiating it properly.

Algebraic Manipulation of Rational Functions

Rational functions are ratios of polynomials, and simplifying or rewriting them can help in integration or differentiation. Recognizing how to express the integrand and the proposed antiderivative in comparable forms is crucial to verify correctness, especially when powers and sums are involved in numerator and denominator.

Checking Antiderivative FormulasRight, or wrong? Give a brief reason why.∫−15(x + 3)² / (x − 2)⁴ dx = ((x + 3)/(x − 2))³ + C