A) \[x{{(\log x)}^{2}}+c\]
B) \[x{{(1+\log x)}^{2}}+c\]
C) \[x[1+{{(\log x)}^{2}}]+c\]
D) None of these
Correct Answer: A
Solution :
\[I=\int_{{}}^{{}}{\log x(\log x+2)\,dx}\] Put \[\log x=t\Rightarrow {{e}^{t}}=x\Rightarrow {{e}^{t}}dt=dx,\] then \[I=\int_{{}}^{{}}{t(t+2){{e}^{t}}dt}={{t}^{2}}.\,{{e}^{t}}+c=x{{(\log x)}^{2}}+c.\]
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