What is the integral of sqrt((x-1)/(x^5)) ?

The integral of sqrt((x-1)/(x^5)) is -(2sqrt(x-1)}{3x^{3/2}}+2/3 sqrt(\frac{x-1)/x)+C

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integral of sqrt((x-1)/(x^5))

\int \frac{x - 1}{x ^{5}} d x

- \frac{2 x - 1}{3 x ^{\frac{3}{2}}} + \frac{2}{3} \frac{x - 1}{x} + C

Solution steps

\int \frac{x - 1}{x ^{5}} d x

Simplify

\frac{x - 1}{x ^{5}} : \frac{x - 1}{x ^{5}}

= \int \frac{x - 1}{x ^{5}} d x

Apply Integration By Parts

: - \frac{2 x - 1}{3 x ^{\frac{3}{2}}} - \int - \frac{1}{3 x ^{\frac{3}{2}} x - 1} d x

= - \frac{2 x - 1}{3 x ^{\frac{3}{2}}} - \int - \frac{1}{3 x ^{\frac{3}{2}} x - 1} d x

\int - \frac{1}{3 x ^{\frac{3}{2}} x - 1} d x = - \frac{2}{3} \frac{x - 1}{x}

= - \frac{2 x - 1}{3 x ^{\frac{3}{2}}} - (- \frac{2}{3} \frac{x - 1}{x})

Simplify

= - \frac{2 x - 1}{3 x ^{\frac{3}{2}}} + \frac{2}{3} \frac{x - 1}{x}

Add a constant to the solution

= - \frac{2 x - 1}{3 x ^{\frac{3}{2}}} + \frac{2}{3} \frac{x - 1}{x} + C

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What is the integral of sqrt((x-1)/(x^5)) ?
The integral of sqrt((x-1)/(x^5)) is -(2sqrt(x-1)}{3x^{3/2}}+2/3 sqrt(\frac{x-1)/x)+C