Integral Calculus Question 67
Question: $ \int_{{}}^{{}}{\frac{dx}{x(x^{n}+1)}=} $
[Roorkee 1979]
Options:
A) $ n\log \frac{x^{n}}{x^{n}+1}+c $
B) $ n\log \frac{x^{n}+1}{x^{n}}+c $
C) $ \frac{1}{n}\log \frac{x^{n}}{x^{n}+1}+c $
D) $ \frac{1}{n}\log \frac{x^{n}+1}{x^{n}}+c $
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Answer:
Correct Answer: C
Solution:
Put $ x^{n}=t\Rightarrow n{x^{n-1}}dx=dt $
$ \Rightarrow \frac{nx^{n}}{x},dx=dt\Rightarrow \frac{1}{x},dx=\frac{dt}{nt}, $ then it reduces to $ \int_{{}}^{{}}{\frac{dt}{nt(t+1)}}=\frac{1}{n}[ \int_{{}}^{{}}{\frac{dt}{t(t+1)}} ] $ $ =\frac{1}{n}[ \int_{{}}^{{}}{\frac{1}{t},dt-\int_{{}}^{{}}{\frac{1}{t+1}\ dt}} ]=\frac{1}{n}\log \frac{x^{n}}{x^{n}+1}+c $ .
BETA
