Integral Calculus Question 70
Question: $ \int_{{}}^{{}}{\frac{dx}{x(x^{5}+1)}}= $
[UPSEAT 2004]
Options:
A) $ \frac{1}{5}\log x^{5}(x^{5}+1)+c $
B) $ \frac{1}{5}\log x^{5}( \frac{1+x^{5}}{x^{5}} )+c $
C) $ \frac{1}{5}\log x^{5}( \frac{x^{5}}{x^{5}+1} )+c $
D) None of these
Show Answer
Answer:
Correct Answer: D
Solution:
We have $ I=\int{\frac{dx}{x(x^{5}+1)}}=\int{\frac{dx}{x^{6}( 1+\frac{1}{x^{5}} )}} $ Put $ 1+\frac{1}{x^{5}}=t $
Þ $ \frac{-5}{x^{6}}dx=dt $
Þ $ I=-\frac{1}{5}\int{\frac{dt}{t}=-\frac{1}{5}}\log t+c $ $ I=-\frac{1}{5}\log ( 1+\frac{1}{x^{5}} )+c=-\frac{1}{5}\log ( \frac{x^{5}+1}{x^{5}} )+c $ \ $ I=\frac{1}{5}\log ( \frac{x^{5}}{x^{5}+1} )+c $ .
BETA
