A) \[\log |{{x}^{2}}+1|+\log |{{x}^{2}}+2|+c\]
B) \[-[\log |{{x}^{2}}+1|+\log |{{x}^{2}}+2|+c\]
C) \[\log |{{x}^{2}}+1|-\log |{{x}^{2}}+2|+c\]
D) \[\log |{{x}^{2}}+2|-\log |{{x}^{2}}+1|+c\]
Correct Answer: C
Solution :
Let \[I=\int{\frac{2x\,dx}{({{x}^{2}}+1)({{x}^{2}}+2)}}\] Put \[{{x}^{2}}=t\] \[\Rightarrow \] \[2xdx=dt\] \[\therefore \] \[I=\int{\frac{dt}{(t+1)(t+2)}}\] \[=\int{\left( \frac{1}{t+1}-\frac{1}{t+2} \right)}dt\] (By partial fraction) \[I=\log |t+1|-\log |t+2|+c\] \[\Rightarrow \] \[I=\log |{{x}^{2}}+1|-\log |{{x}^{2}}+2|+c\] \[(\because t={{x}^{2}})\]
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