A) \[\frac{1}{4}log\left( {{x}^{4}}+1 \right)+c\]
B) \[log\left( 1+{{x}^{4}} \right)+c\]
C) \[-log\left( {{x}^{4}}+1 \right)+c\]
D) \[-\frac{1}{4}log\left( 1+{{x}^{4}} \right)+c\]
Correct Answer: A
Solution :
[a] \[I={{\int{e}}^{3\log \frac{x}{{{x}^{4}}+1}}}\] \[I={{\int{e}}^{\log \frac{{{x}^{3}}}{1+{{x}^{4}}}}}.dx\] \[I=\int{\frac{{{x}^{3}}}{1+{{x}^{4}}}}.dx\] Let \[z=1+{{x}^{4}}\] \[dz=4{{x}^{3}}dx\] Now \[I=\frac{1}{4}\int{\frac{1}{z}.dz}=\frac{1}{4}.\log z=\frac{1}{4}.\log (1+{{x}^{4}})+c\] Hence, option [a] is correct.
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