Evaluate ∫ x log x dx.

Question 80

MediumOriginal question

Evaluate $\int x \log x\,dx$ .

\frac{x^{2} \left(2 \log x - 1\right)}{4} + C

Correct answerYour answer

\frac{x^{2} \left(2 \log x + 1\right)}{4} + C

Correct answerYour answer

\frac{x^{2} \log x}{2} + C

Correct answerYour answer

- \frac{x^{2} \left(2 \log x - 1\right)}{4} + C

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

By parts, $\int u\,dv = uv - \int v\,du$ , with $u = \log x$ and $dv = x\,dx$ , so $v = \frac{x^{2}}{2}$ .
$\frac{x^{2} \log x}{2} - \int \frac{x}{2}\,dx = \frac{x^{2} \left(2 \log x - 1\right)}{4} + C$ .

Worth noting

Adding the second integral instead of subtracting it is the slip one of the options is built on.

Ten questions from Indefinite Integrals, at your own pace.