Question

Evaluate \[ \int (\log 2x)^3 \, dx = ? \]

• A. \(x \left[(\log 2x)^3 - 3 (\log 2x)^2 + 6 (\log 2x) - 6 \right] + c\)
• B. \(\frac{x}{4} \left[4(\log 2x)^3 - 6 (\log 2x)^2 + 6 (\log 2x) - 3 \right] + c\)
• C. \(\frac{x}{2} \left[(\log 2x)^3 - 3 (\log 2x)^2 + 3 (\log 2x) - 6\right] + c\)
• D. \(x \left[(\log 2x)^3 - 6 (\log 2x)^2 + 18 (\log 2x) - 54\right] + c\)

Hint:

Apply integration by parts for powers of logarithmic functions.

Answer 1

The Correct Option is A

Use integration by parts with substitution \(t = \log 2x\). Repeatedly integrate powers of \(t\), and substitute back to get \[ \int (\log 2x)^3 dx = x \left[(\log 2x)^3 - 3 (\log 2x)^2 + 6 (\log 2x) - 6\right] + c. \]