Question

Evaluate $\int \frac{7dx}{x(x^7-1)}$

Answer 1

$\int \frac{7dx}{x(x^7-1)}$

$=\int \frac{7dx}{x^8(1-\frac{1}{x^7})}$

$=\int \frac{\left(\frac{7}{x^8}\right)dx}{(1-\frac{1}{x^7})}$

Let $1-\frac{1}{x^7}=t\Rightarrow \frac{7}{x^8}dx=dt$

Therefore, the integral become-

$\int \frac{dt}{t}$

$=\log t+C$

Substituting the value of $t$ in above equation, we get

$=\log (1-\frac{1}{x^7})+C$

Therefore,

$\int \frac{7dx}{x(x^7-1)}=\log (1-\frac{1}{x^7})+C$