A) \[\frac{1}{2}{{x}^{4}}{{e}^{{{x}^{2}}}}-{{x}^{2}}{{e}^{{{x}^{2}}}}+{{e}^{{{x}^{2}}}}+c\]
B) \[\frac{1}{2}{{x}^{4}}{{e}^{{{x}^{2}}}}+{{x}^{2}}{{e}^{{{x}^{2}}}}+{{e}^{{{x}^{2}}}}+c\]
C) \[\frac{1}{2}{{x}^{4}}{{e}^{{{x}^{2}}}}-{{x}^{2}}{{e}^{{{x}^{2}}}}-{{e}^{{{x}^{2}}}}+c\]
D) None of these
Correct Answer: A
Solution :
Put \[{{x}^{2}}=t\Rightarrow 2x\,dx=dt,\] then \[\int_{{}}^{{}}{{{x}^{5}}{{e}^{{{x}^{2}}}}dx}=\frac{1}{2}\int_{{}}^{{}}{{{t}^{2}}{{e}^{t}}dt}=\frac{1}{2}\left[ {{e}^{t}}{{t}^{2}}-2\int_{{}}^{{}}{t{{e}^{t}}dt} \right]+c\] \[=\frac{{{t}^{2}}{{e}^{t}}}{2}-\left[ t{{e}^{t}}-{{e}^{t}} \right]+c=\frac{1}{2}{{x}^{4}}{{e}^{{{x}^{2}}}}-{{x}^{2}}{{e}^{{{x}^{2}}}}+{{e}^{{{x}^{2}}}}+c.\]
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