A) \[\log x-f(x)+c\]
B) \[f(x)+\log x+c\]
C) \[f(x)-\log x+c\]
D) None of these
Correct Answer: A
Solution :
\[\frac{{{x}^{4}}dx}{x+{{x}^{5}}}=\int{\frac{({{x}^{4}}+1)dx}{x+{{x}^{5}}}}\]\[=\int{\frac{({{x}^{4}}+1)dx}{x+{{x}^{5}}}}\]\[-\int{\frac{dx}{x+{{x}^{5}}}}\] \[=\int{\frac{({{x}^{4}}+1)dx}{x(1+{{x}^{4}})}}-\int{\frac{dx}{x({{x}^{4}}+1)}}\]\[=\int{\frac{dx}{x}}-\int{\frac{dx}{x+{{x}^{5}}}}\] \[=\log x-f(x)-{{c}_{2}}+{{c}_{1}}=\log x-f(x)+c\] Where \[{{c}_{1}}-{{c}_{2}}=c=\]a new constant.
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