143.
a. Show that ∫ ln(x) dx = x ln(x) − x + C.

검증된 단계별 안내

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

Choose \(u = \ln(x)\) and \(dv = dx\). Then, compute \(du = \frac{1}{x} dx\) and \(v = x\).

Apply the integration by parts formula: \(\int \ln(x) \, dx = x \ln(x) - \int x \cdot \frac{1}{x} \, dx\).

Simplify the integral: \(\int x \cdot \frac{1}{x} \, dx = \int 1 \, dx = x\).

Combine the results to write the integral as \(x \ln(x) - x + C\), where \(C\) is the constant of integration.

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.

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. Choosing appropriate u and dv simplifies the integral, as in integrating ln(x) by setting u = ln(x) and dv = dx.

Properties of the Natural Logarithm Function

The natural logarithm function, ln(x), is defined for x > 0 and has the derivative d/dx [ln(x)] = 1/x. Understanding its domain and derivative is essential when integrating ln(x), as it guides the choice of functions in integration by parts and ensures the integral is correctly evaluated.

Constant of Integration

When computing indefinite integrals, the result includes an arbitrary constant C because differentiation of a constant is zero. This constant accounts for all possible antiderivatives, ensuring the general solution to the integral is complete.

143.a. Show that ∫ ln(x) dx = x ln(x) − x + C.