If \(\int f(x)\,dx = x^2\log x+C\) for \(x \gt 0\), then \(f(x)\) is equal to
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(2x\log x+x\)
Integration is the inverse process of differentiation, so \(f(x)\) is the derivative of \(x^2\log x\).
By the product rule:
\( \displaystyle f(x) = \frac{d}{dx}\left(x^2\log x\right) = 2x\log x+x^2\cdot\frac{1}{x} \)
\( f(x) = 2x\log x+x \)
Why the other options are incorrect
Option (1) \(2x\log x\) — Incorrect. It differentiates only \(x^2\) and leaves out the term \(x^2\cdot\frac{1}{x} = x\).
Option (2) \(x\left(2\log x-1\right)\) — Incorrect. The second term of the product rule is \(+x\), not \(-x\).
Option (4) \(\frac{x^3}{3}\log x-\frac{x^3}{9}\) — Incorrect. This is the integral of \(x^2\log x\); the question needs the derivative.