A) \[\frac{1}{2}\log \left( {{x}^{2}}+1 \right)+C\]
B) \[\frac{1}{2}\log \left| {{x}^{2}}+1 \right|+C\]
C) \[\log \left( {{x}^{2}}+1 \right)+C\]
D) \[\log \left| {{x}^{2}}-1 \right|+C\]
Correct Answer: A
Solution :
Let \[I=\int_{{}}^{{}}{\frac{{{x}^{2}}-x}{{{x}^{3}}-{{x}^{2}}+x-1}}dx\] \[=\int_{{}}^{{}}{\frac{x\left( x- \right)}{{{x}^{2}}\left( x-1 \right)+\left( x-1 \right)}}dx=\int_{{}}^{{}}{\frac{x\,dx}{{{x}^{2}}+1}}\] \[=\frac{1}{2}\int_{{}}^{{}}{\frac{2xdx}{\left( {{x}^{2}}+1 \right)}}\] Let\[{{x}^{2}}+1=t\Rightarrow 2xdx=dt\] \[\therefore \]\[I=\frac{1}{2}\int_{{}}^{{}}{\frac{dt}{t}=\frac{1}{2}}\log t+c\] \[=\frac{1}{2}\log \left( {{x}^{2}}+1 \right)+c\]where 'c' is the constant of integration.
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