Question

Evaluate the integral: \[ \int (\log x)^3 \, dx \]

• A. \( ( \log x)^3 - 3 (\log x)^2 + 6 \log x - 6 + c \)
• B. \( x [ (\log x)^3 - 3 (\log x)^2 + 6 \log x - 6 ] + c \)
• C. \( (x \log x)^3 - 3 (x \log x)^2 + 6x \log x - 6x + c \)
• D. \( x [ (\log x)^3 - 3 (\log x)^2 + 6 \log x - 6 ] + c \) \bigskip

Hint:

When faced with integrals involving logarithmic functions raised to a power, use integration by parts. It helps to break down the powers step by step, making the integral simpler to solve.

Answer 1

The Correct Option is B

Step 1: Integration by Parts To solve this integral, use integration by parts. Let: \[ u = (\log x)^3 \quad \text{and} \quad dv = dx. \] Then: \[ du = 3 (\log x)^2 \cdot \frac{1}{x} dx \quad \text{and} \quad v = x. \] By the formula for integration by parts, \( \int u \, dv = uv - \int v \, du \), we get: \[ \int (\log x)^3 \, dx = x [ (\log x)^3 - 3 (\log x)^2 + 6 \log x - 6 ] + c. \] \bigskip