What is the integral of (sqrt(x^2-2x))/((x-1)^3) ?

The integral of (sqrt(x^2-2x))/((x-1)^3) is -(sqrt(x^2-2x))/(2(x-1)^2)+1/2 arctan(sqrt(x^2-2x))+C

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integral of (sqrt(x^2-2x))/((x-1)^3)

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

- \frac{x ^{2} - 2 x}{2 ( x - 1 ) ^{2}} + \frac{1}{2} arctan (x^{2} - 2 x) + C

Solution steps

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

Apply Integration By Parts

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

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

\int - \frac{1}{2 ( x - 1 ) x ^{2} - 2 x} d x = - \frac{1}{2} arctan (x^{2} - 2 x)

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

Simplify

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

Add a constant to the solution

= - \frac{x ^{2} - 2 x}{2 ( x - 1 ) ^{2}} + \frac{1}{2} arctan (x^{2} - 2 x) + C

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What is the integral of (sqrt(x^2-2x))/((x-1)^3) ?
The integral of (sqrt(x^2-2x))/((x-1)^3) is -(sqrt(x^2-2x))/(2(x-1)^2)+1/2 arctan(sqrt(x^2-2x))+C