A) \[{{e}^{x}}.{{x}^{5}}+c\]
B) \[{{e}^{x}}.{{x}^{5}}+{{e}^{x}}+c\]
C) \[{{e}^{x+1}}.{{x}^{5}}+c\]
D) \[5{{x}^{4}}.{{e}^{x}}+c\]
Correct Answer: B
Solution :
Let \[I=\int{{{e}^{x}}({{x}^{5}}+5{{x}^{4}}+1)dx}\] \[=\int{{{e}^{x}}{{x}^{5}}dx+5\int{{{e}^{x}}{{x}^{4}}dx}+\int{{{e}^{x}}dx}}\] \[={{x}^{5}}{{e}^{x}}-\int{5{{x}^{4}}{{e}^{x}}\,\,\,dx+5\int{{{e}^{x}}\,{{x}^{4}}\,dx+{{e}^{x}}}}\] \[={{x}^{5}}{{e}^{x}}+{{e}^{x}}+c={{e}^{x}}({{x}^{5}}+1)+c\]
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