A) \[{{(1+{{x}^{4}})}^{\frac{1}{2}}}+c\]
B) \[-{{(1+{{x}^{4}})}^{\frac{1}{2}}}+c\]
C) \[\frac{1}{2}{{(1+{{x}^{4}})}^{\frac{1}{2}}}+c\]
D) \[-\frac{1}{2}{{(1+{{x}^{4}})}^{\frac{1}{2}}}+c\]
Correct Answer: C
Solution :
\[\int_{{}}^{{}}{\frac{{{x}^{3}}}{\sqrt{1+{{x}^{4}}}}\,dx}=\frac{1}{4}\int_{{}}^{{}}{\frac{4{{x}^{3}}}{\sqrt{1+{{x}^{4}}}}\,dx}\] \[(\text{Put}\,1+{{x}^{4}}=t)\] \[=\frac{1}{4}\int_{{}}^{{}}{\frac{dt}{{{t}^{1/2}}}}=\frac{1}{4}\frac{{{t}^{-\frac{1}{2}+1}}}{-\frac{1}{2}+1}=\frac{1}{2}\sqrt{t}=\frac{1}{2}\sqrt{1+{{x}^{4}}}+c.\]
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