Integral Calculus Question 8
Question: If $ \int{\frac{1}{x+x^{5}}dx=f(x)+c} $ , then the value of $ \int{\frac{x^{4}}{x+x^{5}}dx} $ is
[DCE 2005]
Options:
A) $ \log x-f(x)+c $
B) $ f(x)+\log x+c $
C) $ f(x)-\log x+c $
D) None of these
Show Answer
Answer:
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.
BETA
