The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (2 ln(z³)) / (16z) dz

검증된 단계별 안내

Start by simplifying the integrand. Use the logarithm power rule: \(\ln(z^3) = 3 \ln(z)\), so rewrite the integral as \(\int \frac{2 \cdot 3 \ln(z)}{16z} \, dz\).

Simplify the constants in the integrand: \(\frac{2 \cdot 3}{16} = \frac{6}{16} = \frac{3}{8}\). The integral becomes \(\int \frac{3}{8} \cdot \frac{\ln(z)}{z} \, dz\).

Factor out the constant \(\frac{3}{8}\) from the integral: \(\frac{3}{8} \int \frac{\ln(z)}{z} \, dz\).

Recognize that the integral \(\int \frac{\ln(z)}{z} \, dz\) can be solved using substitution. Let \(u = \ln(z)\), then \(du = \frac{1}{z} dz\), which matches the integrand's differential part.

Rewrite the integral in terms of \(u\): \(\frac{3}{8} \int u \, du\). Then integrate \(u\) with respect to \(u\) using the power rule for 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.

Properties of Logarithms

Understanding logarithm properties, such as ln(a^b) = b ln(a), allows simplification of expressions involving logarithms. This is essential for rewriting ln(z³) as 3 ln(z), making the integral easier to handle.

Integration by Substitution

Integration by substitution involves changing variables to simplify an integral. Recognizing parts of the integrand as derivatives of a function helps to choose an appropriate substitution, streamlining the integration process.

Basic Integration Rules

Familiarity with basic integration formulas, such as ∫ ln(x)/x dx or power rule integrals, is crucial. These rules help evaluate integrals after simplification or substitution, enabling the calculation of antiderivatives efficiently.

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.∫ (2 ln(z³)) / (16z) dz