Question

The value of $\int \frac{dx}{x^{13}(x^6-1)}$ is

• A. $\log \left(|\frac{x^6-1}{x^6}|\right)+x^{-6}+x^{-12}+c$
• B. $\frac{1}{6}\log (|\frac{x^6-1}{x^6}|)+\frac{1}{6}{x}^{-6}+\frac{1}{4}{x}^{-12}+c$
• C. $\frac{1}{6}(\log |\frac{x^6-1}{x^6}|+x^{-6}+\frac{1}{2}{x}^{-12})+c$
• D. $\frac{1}{6}\log (|\frac{x^6-1}{x^6}|)+c$

Answer 1

The correct option is C $\frac{1}{6}(\log |\frac{x^6-1}{x^6}|+x^{-6}+\frac{1}{2}{x}^{-12})+c$

$\int \frac{1}{x^7\left(x^{12}\right)}{[1-1/x^6]}^{dx}$

$1/x6=t$

$\frac{-6}{x^7}dx=dt$

$1/x^{7}dx=\frac{dt}{-6}$

$\int \frac{t^{2}dt}{-6(1-t)}=+1/6\int \frac{t^2}{(t-1)}\cdot dt$

$=1/6\int \frac{(t^2-1)+1}{t-1}dt$

$=1/6\int (t+1)+\frac{1}{(t-1)}dt$

$=1/6[[\frac{t^2}{2}+t]+log(t-1\left)\right]+c$

$1/6[\frac{x^{-12}}{2}+x^{-6}+log\left|\frac{x^6-1}{x^6}\right|]+c$