Question

$\int\frac{dx}{x\left(x^{n}+1\right)}$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 these

Answer 1

Answer: (A)

$ I=\int\frac{dx}{x\left(x^{n}+1\right)}=\int\frac{x^{n-1}}{x^{2}\left(x^{n}+1\right) }dx$
Putting $x^{n} = t$ so that $nx^{n-1}dx = dt, i.e.$
we get $ x^{n-1}dx = \frac{1}{n}dt$
$I = \int\frac{\frac{1}{n}dt}{t\left(t-1\right)} = \frac{1}{n}\int\left(\frac{1}{t}-\frac{1}{t+1}\right)dt$
$=\frac{1}{n}\left(log t-log\left (t+1\right)\right)+C$
$=\frac{1}{n} log \left(\frac{x^{n}}{x^{n}+1}\right)+C$