Question

Integrate (sqrt((x-1)/x^5)) using substitution (no trig substitution) ...?

Answer 1

√ (( x - 1) / x^5) = 1/√x^4 · √(1 - 1/x )= 1/x² · √( 1 - 1/x )
Substitution:
u = 1 - 1/x
d u = 1/x² dx
$\int { \frac{1}{ x^{2} } * \sqrt{1-1/x} } \, dx= \\ = \int { \sqrt{u} } \, du= \\ =2/3u \sqrt{u}= \\ =2/3(1-1/x) \sqrt{1-1/x}+C$