A) \[\frac{1}{\sqrt{\frac{1}{{{x}^{2}}}-{{x}^{4}}}}+c\]
B) \[\frac{2x}{\sqrt{1+{{x}^{6}}}}+c\]
C) \[\frac{{{x}^{2}}+x}{\sqrt{1-{{x}^{6}}}}+c\]
D) None of these
Correct Answer: A
Solution :
[a] \[I=\int{\frac{1+2{{x}^{6}}}{{{(1-{{x}^{6}})}^{3/2}}}}\,\,dx=\int{\frac{\frac{1}{{{x}^{3}}}+2{{x}^{3}}}{{{\left( \frac{1}{{{x}^{2}}}-{{x}^{4}} \right)}^{\frac{3}{2}}}}}\,\,dx\] Put \[\frac{1}{{{x}^{2}}}-{{x}^{4}}=t\] \[\therefore \,\,\,\,\,\,\,I=\frac{-1}{2}\int{\frac{dt}{{{t}^{3/2}}}}={{t}^{-1/2}}+c=\frac{1}{\sqrt{\frac{1}{{{x}^{2}}}-{{x}^{4}}}}+c\]
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