Integral Calculus Question 76
Question: $ \int \frac{dx}{x(x^{n}+1)} $ is equal to
Options:
A) $ \frac{1}{n}\log ( \frac{x^{n}}{x^{n}+1} )+c $
B) $ \frac{1}{n}\log ( \frac{x^{n}+1}{x^{n}} )+c $
C) $ \log ( \frac{x^{n}}{x^{n}+1} )+c $
D) none of these
Show Answer
Answer:
Correct Answer: A
Solution:
[a] $ I=\int{\frac{dx}{x(x^{n}+1)}=\int{\frac{{x^{n-1}}}{x^{n}(x^{n}+1)}dx}} $ Putting $ x^{n}=t $ so that $ n{x^{n-1}}dx=dt. $ i.e., We get $ {x^{n-1}}dx=\frac{1}{n}dt $ $ I=\int{\frac{\frac{1}{n}dt}{t(t+1)}}=\frac{1}{n}\int{( \frac{1}{t}-\frac{1}{t+1} )dt} $ $ =\frac{1}{n}(\log t-\log ,(t+1))+C $ $ =\frac{1}{n}\log ( \frac{x^{n}}{x^{n}+1} )+C $
BETA
