Question

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

• A. $n$ $(log{x}^n$-$log(x^n+1))+C$
• B. $\frac{1}{n}$ $(log{x}^n$+$log(x^n+1))+C$
• C. $\frac{1}{n}$ $(log{x}^n$-$log(x^n+1))+C$
• D. $n$ $(log{x}^n$+$log(x^n+1))+C$

Answer 1

Answer: (C)

$\frac{1}{n}$ $(log{x}^n$-$log(x^n+1))+C$

$I=\int \left(\frac{1}{x(x^n+1)}\right)dxI=\int \left(\frac{1}{x^n(x^n+1)}\right)dxlet{x}^n=tthenn{x}^{n-1}dx=dt\therefore I=\left(\frac{1}{n}\right)\int \left(\frac{1}{t(t+1)}\right)dx=\left(\frac{1}{n}\right)\int \left(\right(\frac{1}{t})-(\frac{1}{t+1}\left)\right)dx=\left(\frac{1}{n}\right)[logt-log(t+1\left)\right]+c=\left(\frac{1}{n}\right)[log{x}^n-log(x^n+1\left)\right]+c$