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

The integral of (xsqrt(1-x))/(sqrt(2-x)) is

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integral of (xsqrt(1-x))/(sqrt(2-x))

\int \frac{x 1 - x}{2 - x} d x

- 2 x 1 - x 2 - x - 2 (\frac{3}{4} (- x + 2)^{\frac{3}{2}} - x + 1 - \frac{7}{8} - x + 2 - x + 1 - \frac{7}{8} ln - x + 2 + - x + 1) + C

Solution steps

\int \frac{x 1 - x}{2 - x} d x

Factor

1 - x : - (- 1 + x)

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

Apply Integration By Parts

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

\int - \frac{( - 3 x + 2 ) 2 - x}{1 - x} d x = 2 (3 (\frac{1}{4} (2 - x)^{\frac{3}{2}} - x + 1 + \frac{3}{8} (2 - x - x + 1 + ln - x + 1 + 2 - x)) - 4 (\frac{1}{2} 2 - x - x + 1 + \frac{1}{2} ln - x + 1 + 2 - x))

= - 2 x 1 - x 2 - x - 2 (3 (\frac{1}{4} (2 - x)^{\frac{3}{2}} - x + 1 + \frac{3}{8} (2 - x - x + 1 + ln - x + 1 + 2 - x)) - 4 (\frac{1}{2} 2 - x - x + 1 + \frac{1}{2} ln - x + 1 + 2 - x))

Expand

3 (\frac{1}{4} (2 - x)^{\frac{3}{2}} - x + 1 + \frac{3}{8} (2 - x - x + 1 + ln - x + 1 + 2 - x)) - 4 (\frac{1}{2} 2 - x - x + 1 + \frac{1}{2} ln - x + 1 + 2 - x) : \frac{3}{4} (- x + 2)^{\frac{3}{2}} - x + 1 - \frac{7}{8} - x + 2 - x + 1 - \frac{7}{8} ln - x + 2 + - x + 1

= - 2 x 1 - x 2 - x - 2 (\frac{3}{4} (- x + 2)^{\frac{3}{2}} - x + 1 - \frac{7}{8} - x + 2 - x + 1 - \frac{7}{8} ln - x + 2 + - x + 1)

Add a constant to the solution

= - 2 x 1 - x 2 - x - 2 (\frac{3}{4} (- x + 2)^{\frac{3}{2}} - x + 1 - \frac{7}{8} - x + 2 - x + 1 - \frac{7}{8} ln - x + 2 + - x + 1) + C

What is the integral of (xsqrt(1-x))/(sqrt(2-x)) ?
The integral of (xsqrt(1-x))/(sqrt(2-x)) is