A) \[\frac{2}{9}{{(1+{{x}^{3}})}^{\frac{5}{2}}}+\frac{2}{3}{{(1+{{x}^{3}})}^{\frac{3}{2}}}+c\]
B) \[\log |\sqrt{x}+\sqrt{1+{{x}^{3}}}|+\,c\]
C) \[\log |\sqrt{x}-\sqrt{1+{{x}^{3}}}|+\,c\]
D) \[\frac{2}{9}{{(1+{{x}^{3}})}^{\frac{3}{2}}}-\frac{2}{3}{{(1+{{x}^{3}})}^{\frac{1}{2}}}+c\]
Correct Answer: D
Solution :
Given, \[I=\int{\frac{{{x}^{5}}}{\sqrt{1+{{x}^{3}}}}dx}\] Let \[1+{{x}^{3}}=t\] \[\Rightarrow \] \[3{{x}^{2}}dx=dt\] \[\Rightarrow \] \[{{x}^{2}}dx=\frac{dt}{3}\] \[\therefore \]\[I=\int{\frac{(t-1)}{\sqrt{t}}}\cdot \frac{dt}{3}=\frac{1}{3}\int{(\sqrt{t}-{{t}^{-1/2}})dt}\] \[=\frac{1}{3}\left[ \frac{2{{t}^{3/2}}}{3}-2{{t}^{1/2}} \right]+c\] \[=\frac{2}{9}{{(1+{{x}^{3}})}^{3/2}}-\frac{2}{3}{{(1+{{x}^{3}})}^{1/2}}+c\]
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