Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ (ln x)³/x dx

검증된 단계별 안내

Recognize that the integral is \( \int \frac{(\ln x)^3}{x} \, dx \). Notice that the integrand involves \( (\ln x)^3 \) divided by \( x \), which suggests a substitution related to \( \ln x \).

Use the substitution \( u = \ln x \). Then, the differential \( du = \frac{1}{x} dx \), which means \( dx = x \, du \). Since \( \frac{1}{x} dx = du \), the integral can be rewritten in terms of \( u \).

Rewrite the integral using the substitution: \( \int (\ln x)^3 \cdot \frac{1}{x} dx = \int u^3 \, du \). This simplifies the integral to a basic power function of \( u \).

Integrate \( u^3 \) with respect to \( u \) using the power rule for integration: \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \), where here \( n = 3 \).

After integrating, substitute back \( u = \ln x \) to express the answer in terms of \( x \). This completes the evaluation of the integral.

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주요 개념

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

Integration by Substitution

Integration by substitution simplifies integrals by changing variables to make the integral easier to solve. For example, setting u = ln(x) transforms the integral into a polynomial form in terms of u, which is easier to integrate.

Properties of Logarithmic Functions

Understanding the natural logarithm function ln(x) and its derivative is crucial. Since d/dx [ln(x)] = 1/x, this relationship helps identify substitution candidates and simplifies integrals involving ln(x).

Integration of Polynomial Functions

Once substitution reduces the integral to a polynomial form, integrating powers of a variable (like u³) involves applying the power rule for integration: ∫u^n du = u^(n+1)/(n+1) + C, which is straightforward and essential for solving the problem.

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.∫ (ln x)³/x dx