A) \[\left( x-\frac{{{x}^{3}}}{3} \right)\log x-\left( x-\frac{{{x}^{3}}}{9} \right)+c\]
B) \[\left( x-\frac{{{x}^{3}}}{3} \right)\log x+\left( x-\frac{{{x}^{3}}}{9} \right)+c\]
C) \[\left( x+\frac{{{x}^{3}}}{3} \right)\log x+\left( x+\frac{{{x}^{3}}}{9} \right)+c\]
D) None of these
Correct Answer: A
Solution :
\[\int_{{}}^{{}}{(1-{{x}^{2}})\log x\,dx}=\int_{{}}^{{}}{\log x\,dx}-\int_{{}}^{{}}{{{x}^{2}}\log x\,dx}\] \[=x(\log x-1)-\frac{{{x}^{3}}\log x}{3}+\frac{{{x}^{3}}}{9}+c\] \[=\left( x-\frac{{{x}^{3}}}{3} \right)\log x-\left( x-\frac{{{x}^{3}}}{9} \right)+c.\]
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