A) \[\frac{2}{9}{{(1+{{x}^{3}})}^{3/2}}+c\]
B) \[\frac{2}{9}{{(1+{{x}^{3}})}^{3/2}}+\frac{2}{3}{{(11-{{x}^{3}})}^{1/2}}+c\]
C) \[\frac{2}{9}{{(1+{{x}^{3}})}^{3/2}}-\frac{2}{3}{{(1+{{x}^{3}})}^{1/2}}+c\]
D) none of these
Correct Answer: C
Solution :
Let \[I=\int_{{}}^{{}}{\frac{{{x}^{5}}}{\sqrt{1+{{x}^{3}}}}dx}\] Put \[1+{{x}^{3}}={{t}^{2}}\Rightarrow 3{{x}^{2}}dx=2t\,dt\] \[\therefore \] \[I=\frac{2}{3}\int_{{}}^{{}}{\frac{({{t}^{2}}-1)t\,dt}{t}}\] \[=\frac{2}{3}\int_{{}}^{{}}{({{t}^{2}}-1)\,dt}\] \[=\frac{2}{3}\left[ \frac{{{t}^{3}}}{3}-t \right]+c\] \[=\frac{2}{3}\left[ \frac{{{(1+{{x}^{3}})}^{3/2}}}{3}-{{(1+{{x}^{3}})}^{1/2}} \right]+c\] \[=\frac{2}{9}{{(1+{{x}^{3}})}^{3/2}}-\frac{2}{3}{{(1+{{x}^{3}})}^{1/2}}+c\]
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