A) \[\frac{1}{3}\left[ {{x}^{3}}\psi ({{x}^{3}})-\int{{{x}^{2}}\psi ({{x}^{3}})dx} \right]+C\]
B) \[\frac{1}{3}{{x}^{3}}\psi ({{x}^{3}})-3\int{{{x}^{3}}\psi ({{x}^{3}})dx}+C\]
C) \[\frac{1}{3}{{x}^{3}}\psi ({{x}^{3}})-\int{{{x}^{3}}\psi ({{x}^{3}})dx}+C\]
D) \[\frac{1}{3}\left[ {{x}^{3}}\psi ({{x}^{3}})-\int{{{x}^{3}}\psi ({{x}^{3}})dx} \right]+C\]
Correct Answer: C
Solution :
\[\int{f(x)\text{ }dx=\psi (x),}\text{ }then\int{{{x}^{5}}f({{x}^{3}})dx}\] \[I=\int{{{x}^{3}}.{{x}^{2}}f{{(x)}^{3}}dx}\] Let \[{{x}^{3}}=t\] \[3{{x}^{2}}\text{ }dx=dt\] \[I=\int{t\,f(t)\frac{dt}{3}}\] \[I=\frac{1}{3}\left[ t\int{f(t)dt-\int{f(t)dt}} \right]\] \[I=\frac{{{x}^{3}}}{3}\psi ({{x}^{3}})-\frac{1}{3}\int{\frac{d}{dt}}(t)\int{f(t)dt}+c\] \[I=\frac{{{x}^{3}}}{3}\psi ({{x}^{3}})-\frac{1}{3}\int{3{{x}^{2}}\psi ({{x}^{3}})dx}+c\] \[I=\frac{{{x}^{3}}}{3}\psi ({{x}^{3}})-{{x}^{2}}\psi ({{x}^{3}})dx+c\]
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