A) \[{{(\log x)}^{2}}\]
B) \[\frac{1}{2}{{(\log x)}^{2}}\]
C) \[\frac{\log {{x}^{2}}}{2}\]
D) none of these
Correct Answer: A
Solution :
Let \[I=\int_{1}^{x}{\frac{\log {{x}^{2}}}{x}}dx\] \[=\int_{1}^{x}{\frac{2\log x}{x}}dx\] Put \[\log x=t\Rightarrow \frac{1}{x}dx=dt\] \[\therefore \] \[I=2\int_{0}^{\log x}{t\,dt\,=[{{t}^{2}}]_{0}^{\log x}}\] \[={{(\log x)}^{2}}\]
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