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Question: Answered & Verified by Expert
$\int \frac{\log \sqrt{x}}{3 x} d x$ is equal to
MathematicsIndefinite IntegrationWBJEEWBJEE 2016
Options:
  • A $\frac{1}{3}(\log \sqrt{x})^{2}+C$
  • B $\frac{2}{3}(\log \sqrt{x})^{2}+C$
  • C $\frac{2}{3}(\log x)^{2}+C$
  • D $\frac{1}{3}(\log x)^{2}+C$
Solution:
2376 Upvotes Verified Answer
The correct answer is: $\frac{1}{3}(\log \sqrt{x})^{2}+C$
Let $I=\int \frac{\log \sqrt{x}}{3 x} d x$
Aqain, let $\log \sqrt{x}=z \Rightarrow \frac{1}{2 x} d x=d z$
$I=\int \frac{2 z}{3} d z=\frac{2}{3} \int z d z$
$=\frac{2}{3} \cdot \frac{z^{2}}{2}+C=\frac{1}{3} \left(\log \sqrt{x}\right)^{2}+C$

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