A) \[8{{x}^{4}}{{(\log \,\,x)}^{2}}+C\]
B) \[{{x}^{4}}\{8{{(\log \,\,x)}^{2}}-4(\log \,\,x)+1\}+C\]
C) \[{{x}^{4}}\{8{{(\log \,\,x)}^{2}}-4(\log \,\,x)\}+C\]
D) \[{{x}^{3}}\{{{(\log \,\,x)}^{2}}-2\log \,\,x\}+C\]
Correct Answer: B
Solution :
[b] Let \[I=\int{32{{x}^{3}}{{(\log \,\,x)}^{2}}dx}\] \[=32\left\{ {{(log\,\,x)}^{2}}{{\frac{x}{4}}^{4}}-\int{2\log x\frac{1}{x}.\frac{{{x}^{4}}}{4}dx} \right\}\] \[=\frac{32}{4}{{x}^{4}}{{(\log \,\,x)}^{2}}-16\int{{{x}^{3}}\log x\,\,dx}\] \[=8{{x}^{4}}{{(\log x)}^{2}}-4{{x}^{4}}\log x+4\int{{{x}^{3}}dx}\] \[={{x}^{4}}\{8{{(\log \,\,x)}^{2}}-4\,\,\log \,\,x+1\}+C\]
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