Q. $\int_{}^{}\frac{dx}{x(x^{n} + 1)}$ is equal to
A) $\frac{1}{n}\log{\left( \frac{x^{n}}{x^{n} + 1} \right) + c}$
B) $\frac{1}{n}\log{\left( \frac{x^{n} + 1}{x^{n}} \right) + c}$
C) $\log{\left( \frac{x^{n}}{x^{n} + 1} \right) + c}$
D) None of the above
Explanation
(a)
Let $I = \int_{}^{}\frac{dx}{x\left( x^{n} + 1 \right)}$
Put $x^{n} + 1 = t\ \ \ \Rightarrow \ \ \ nx^{n - 1}dx = dt$
$\therefore\ I = \frac{1}{n}\int_{}^{}\frac{dt}{t(t - 1)} = \frac{1}{n}\int_{}^{}\left( \frac{1}{t - 1} - \frac{1}{t} \right)dt$
$= \frac{1}{n}\left\lbrack \log{(t - 1) - \log t} \right\rbrack + c$
$= \frac{1}{n}\log{\left( \frac{t - 1}{t} \right) + c = \frac{1}{n}\log\left( \frac{x^{n}}{x^{n} + 1} \right)} + c$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.