Evaluate the given expression. \int \sqrt{81−(\log ⁡x)^{2}.}}/{x}dx

Correct Answer is : log⁡x281−(log⁡x)2.+812sin−1⁡(log⁡x9)+C

\frac{x}{2}\sqrt{81−(\log ⁡x)^{2}.} + \frac{81}{2} \sin ^{−1}⁡ (\frac{x}{9}) + C

\frac{x}{2}\sqrt{81−(x)^{2}.} + \frac{81}{2} \sin ^{−1}⁡ (\frac{x}{9}) + C

\frac{\log ⁡x}{2}\sqrt{81−(\log ⁡x)^{2}.} + \frac{81}{2} \sin ^{−1}⁡ (\frac{\log ⁡x}{9}) + C

\frac{\log ⁡x}{2}\sqrt{81−(x)^{2}.} + \frac{9}{2} \sin ^{−1}⁡ (\frac{\log ⁡x}{9}) + C

AAI ATC Junior Executive 27 Dec 2023 Shift 1 Paper

Question : 74 of 120

Marks: +1, -0

Evaluate the given expression.

∫ \;{√{81−(\log ⁡x)^{2}.}}/{x}dx

$\;{x}/{2}√{81−(\log ⁡x)^{2}.} + \;{81}/{2} sin \;^{−1}⁡ (\;{x}/{9}) + C$
$\;{x}/{2}√{81−(x)^{2}.} + \;{81}/{2} sin \;^{−1}⁡ (\;{x}/{9}) + C$
$\;{\log ⁡x}/{2}√{81−(\log ⁡x)^{2}.} + \;{81}/{2} sin \;^{−1}⁡ (\;{\log ⁡x}/{9}) + C$
$\;{\log ⁡x}/{2}√{81−(x)^{2}.} + \;{9}/{2} sin \;^{−1}⁡ (\;{\log ⁡x}/{9}) + C$

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