Integral Calculus Question 69
Question: $ \int_{{}}^{{}}{\frac{dx}{x(x^{7}+1)}}= $
[Karnataka CET 2004]
Options:
A) $ \log ( \frac{x^{7}}{x^{7}+1} )+c $
B) $ \frac{1}{7}\log ( \frac{x^{7}}{x^{7}+1} )+c $
C) $ \log ( \frac{x^{7}+1}{x^{7}} )+c $
D) $ \frac{1}{7}\log ( \frac{x^{7}+1}{x^{7}} )+c $
Show Answer
Answer:
Correct Answer: B
Solution:
Given, $ \int_{{}}^{{}}{\frac{dx}{x,(x^{7}+1)}}=\int_{{}}^{{}}{\frac{dx}{x^{8}( 1+\frac{1}{x^{7}} )}} $ Put $ 1+\frac{1}{x^{7}}=t $
Þ $ \frac{-7}{x^{8}}dx=dt $ \ $ I=\frac{-1}{7}\int{\frac{dt}{t}=}\frac{-1}{7}\log t+c $
Þ $ I=-\frac{1}{7}\log ( \frac{x^{7}+1}{x^{7}} )+c $
Þ $ I=\frac{1}{7}\log ( \frac{x^{7}}{x^{7}+1} )+c $ .
BETA
