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

The integral of (sqrt(2x-x^2))/(x^2) is (sqrt(2x-x^2))/x-arcsin(x-1)-6/(tan(1/2 arcsin(x-1))+1)+C

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

\int \frac{2 x - x ^{2}}{x ^{2}} d x

\frac{2 x - x ^{2}}{x} - arcsin (x - 1) - \frac{6}{tan ( \frac{1}{2} arcsin ( x - 1 ) ) + 1} + C

Solution steps

\int \frac{2 x - x ^{2}}{x ^{2}} d x

Apply Integration By Parts

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

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

\int \frac{x - 3}{x - x ^{2} + 2 x} d x = arcsin (x - 1) + \frac{6}{tan ( \frac{1}{2} arcsin ( x - 1 ) ) + 1}

= \frac{2 x - x ^{2}}{x} - (arcsin (x - 1) + \frac{6}{tan ( \frac{1}{2} arcsin ( x - 1 ) ) + 1})

Simplify

= \frac{2 x - x ^{2}}{x} - arcsin (x - 1) - \frac{6}{tan ( \frac{1}{2} arcsin ( x - 1 ) ) + 1}

Add a constant to the solution

= \frac{2 x - x ^{2}}{x} - arcsin (x - 1) - \frac{6}{tan ( \frac{1}{2} arcsin ( x - 1 ) ) + 1} + C

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What is the integral of (sqrt(2x-x^2))/(x^2) ?
The integral of (sqrt(2x-x^2))/(x^2) is (sqrt(2x-x^2))/x-arcsin(x-1)-6/(tan(1/2 arcsin(x-1))+1)+C