\(\displaystyle\int \sqrt{x}\,\log x\,dx,\ x \gt 0\), is equal to (where C is an arbitrary constant)
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{2}{3}x^{3/2}\log x - \frac{4}{9}x^{3/2} + C\)
We use integration by parts, taking log x as the part to differentiate:
\( \displaystyle u = \log x, \quad du = \frac{1}{x}\,dx \)
\( \displaystyle dv = x^{1/2}\,dx, \quad v = \frac{2}{3}x^{3/2} \)
Then
\( \displaystyle \int \sqrt{x}\,\log x\,dx = \frac{2}{3}x^{3/2}\log x - \int \frac{2}{3}x^{3/2} \times \frac{1}{x}\,dx \)
\( \displaystyle = \frac{2}{3}x^{3/2}\log x - \frac{2}{3}\int x^{1/2}\,dx \)
\( \displaystyle = \frac{2}{3}x^{3/2}\log x - \frac{2}{3} \times \frac{2}{3}x^{3/2} + C \)
\( \displaystyle = \frac{2}{3}x^{3/2}\log x - \frac{4}{9}x^{3/2} + C \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{2}{3}x^{3/2}\log x - \frac{2}{3}x^{3/2} + C\) — Incorrect. The last integral \(\int x^{1/2}dx\) has been taken as \(x^{3/2}\); it is \(\frac{2}{3}x^{3/2}\).
Option (2) \(\displaystyle\frac{2}{3}x^{3/2}\log x + \frac{4}{9}x^{3/2} + C\) — Incorrect. The second term of integration by parts is subtracted, not added.
Option (4) \(\displaystyle\frac{3}{2}x^{3/2}\log x - \frac{9}{4}x^{3/2} + C\) — Incorrect. The integral of \(x^{1/2}\) is \(\frac{x^{3/2}}{3/2} = \frac{2}{3}x^{3/2}\); the fraction has been turned upside down.