Question

\(\int x^3logx\,dx =\)

• A. \(\frac{x^4}{16}[4logx-1]+c\)
• B. \(\frac{x^4}{16}[4logx+1]+c\)
• C. \(\frac{x^4}{16}[-4logx-1]+c\)
• D. \(\frac{x^4}{16}[-4logx+1]+c\)

Hint:

Use integration by parts with \(u=\log x\) and \(dv=x^3dx\).

Answer 1

The Correct Option is A

Step 1: Differentiate the candidate
$\dfrac{d}{dx}\left[\dfrac{x^4}{4}\log x-\dfrac{x^4}{16}\right]=x^3\log x+\dfrac{x^3}{4}-\dfrac{x^3}{4}=x^3\log x$.

Step 2: Conclude
This matches the integrand, so option (A) is right.

Final Answer:
Option (A) is the antiderivative. \[ \boxed{\dfrac{x^4}{16}\left[4\log x-1\right]+c} \]