A) \[\frac{1}{6}\left[ {{x}^{6}}g({{x}^{6}})-\int{{{x}^{5}}g({{x}^{6}})} \right]+C\]
B) \[\frac{1}{6}{{x}^{6}}g({{x}^{6}})-\int{{{x}^{5}}g({{x}^{6}})dx}+C\]
C) \[\frac{1}{6}\left[ {{x}^{6}}g({{x}^{6}})-5\int{{{x}^{5}}g({{x}^{6}})dx} \right]+C\]
D) None of these
Correct Answer: B
Solution :
[b] \[I=\int{{{x}^{11}}f({{x}^{6}})dx}\] put \[{{x}^{6}}=t\text{ }\Rightarrow 6{{x}^{5}}dx=dt\] \[I=\frac{1}{6}\int{t\,f(t)dt}\] \[=\frac{1}{6}\left[ tg(t)-\int{g(t)dt} \right]\] \[=\frac{1}{6}\left[ {{x}^{6}}g({{x}^{6}})-\int{g({{x}^{6}})d({{x}^{6}})} \right]\] \[=\frac{1}{6}\left[ {{x}^{6}}g({{x}^{6}})-6\int{{{x}^{5}}g({{x}^{6}})}dx \right]\] \[=\frac{1}{6}{{x}^{6}}g({{x}^{6}})-\int{{{x}^{5}}g({{x}^{6}})dx+c}\]
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