A) Substitute \[{{x}^{2}}=t\]
B) Substitute \[(3{{x}^{2}}+5)=t\]
C) Integrate by parts
D) None of these
Correct Answer: B
Solution :
\[\int_{{}}^{{}}{{{x}^{3}}{{e}^{3{{x}^{2}}+5}}dx}\] The simplest way is substituting \[(3{{x}^{2}}+5)=t.\] Put \[t=3{{x}^{2}}+5\Rightarrow dx=\frac{dt}{6x},\] then \[\int_{{}}^{{}}{{{x}^{3}}{{e}^{3{{x}^{2}}+5}}dx}=\frac{1}{6}\int_{{}}^{{}}{\left( \frac{t-5}{3} \right)\text{ }{{e}^{t}}dt}\] \[=\frac{1}{18}\int_{{}}^{{}}{[t{{e}^{t}}-5{{e}^{t}}]dt}=\frac{1}{18}\int_{{}}^{{}}{t{{e}^{t}}dt}-\frac{5}{18}\int_{{}}^{{}}{{{e}^{t}}dt}\] \[=\frac{1}{18}\left[ t{{e}^{t}}-\int_{{}}^{{}}{{{e}^{t}}dt} \right]-\frac{5}{18}\int_{{}}^{{}}{{{e}^{t}}dt}+c\]\[=\frac{1}{18}(t{{e}^{t}})-\frac{1}{3}{{e}^{t}}+c\] \[=\frac{1}{18}(3{{x}^{2}}+5)\,{{e}^{3{{x}^{2}}+5}}-\frac{1}{3}{{e}^{3{{x}^{2}}+5}}+c.\]
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