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