A) \[\frac{{{x}^{2}}{{7}^{x}}}{\log \,7}+2x\frac{{{7}^{x}}}{{{(\log \,7)}^{2}}}+2\frac{{{7}^{x}}}{{{(\log \,7)}^{3}}}+c\]
B) \[\frac{{{x}^{2}}{{7}^{x}}}{\log \,7}-2x\frac{{{7}^{x}}}{{{(\log \,7)}^{2}}}+2\frac{{{7}^{x}}}{{{(\log \,7)}^{3}}}+c\]
C) \[{{x}^{2}}{{7}^{x}}-2x\,\frac{7x}{\log \,7}+2\,\frac{7x}{{{(\log \,7)}^{2}}}+c\]
D) \[\frac{{{x}^{2}}{{7}^{x}}}{{{(\log \,7)}^{2}}}-2x\,\frac{{{7}^{x}}}{{{(\log \,7)}^{3}}}+2\frac{{{7}^{x}}}{{{(\log \,7)}^{4}}}+c\]
Correct Answer: B
Solution :
\[\int{{{x}^{2}}}\,.\,\,{{7}^{x}}\,\,dx\] \[=\frac{{{x}^{2}}{{.7}^{x}}}{\log \,7}-\int{\frac{2x.\,{{7}^{x}}}{log\,7}}\,dx+c\] \[=\frac{{{x}^{2}}{{.7}^{x}}}{\log 7}-\frac{2}{\log 7}[\int{x{{.7}^{x}}\,dx]+c}\]
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