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