A) \[{{e}^{x}}({{x}^{3}}+3{{x}^{2}})\]
B) \[{{x}^{3}}{{e}^{x}}\]
C) \[{{a}^{3}}{{e}^{a}}\]
D) None of these
Correct Answer: B
Solution :
\[f(x)=\int_{a}^{x}{{{t}^{3}}{{e}^{t}}dt=\int_{a}^{0}{{{t}^{3}}.{{e}^{t}}dt+\int_{0}^{x}{{{t}^{3}}{{e}^{t}}\,\,dt}}}\] \[\Rightarrow \frac{df(x)}{dx}=\frac{d}{dx}\left( \int_{a}^{0}{{{t}^{3}}.{{e}^{t}}dt} \right)+\frac{d}{dx}\left( \int_{0}^{x}{{{t}^{3}}.{{e}^{t}}\,dt} \right)={{x}^{3}}{{e}^{x}}\].
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