A) \[\frac{1}{2}\log \frac{(1-{{x}^{2}})}{{{x}^{2}}}+c\]
B) \[\log \frac{(1-x)}{x(1+x)}+c\]
C) \[\log x(1-{{x}^{2}})+c\]
D) \[\frac{1}{2}\log \frac{{{x}^{2}}}{(1-{{x}^{2}})}+c\]
Correct Answer: D
Solution :
\[\int_{{}}^{{}}{\frac{1}{x-{{x}^{3}}}\,dx=\int_{{}}^{{}}{\frac{1}{x(1+x)(1-x)}\,dx}}\] \[=\frac{1}{2}\int_{{}}^{{}}{\left( \frac{2}{x}-\frac{1}{1+x}+\frac{1}{1-x} \right)\,dx}\] \[=\frac{1}{2}[2\log x-\log (1+x)-\log (1-x)]=\frac{1}{2}\log \frac{{{x}^{2}}}{(1-{{x}^{2}})}+c\].
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