What is the integral of (x(x-1))/(sqrt(1-x)\sqrt{2-x)} ?

The integral of (x(x-1))/(sqrt(1-x)\sqrt{2-x)} is

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integral of (x(x-1))/(sqrt(1-x)\sqrt{2-x)}

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

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

Solution steps

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

Simplify

\frac{x ( x - 1 )}{1 - x 2 - x} : - \frac{x 1 - x}{2 - x}

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \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

= - (- 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)))

Simplify

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

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

Add a constant to the solution

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

\int yz d x

cos (e^{t a n (2 x)})

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\int_{0}^{3} π (\frac{1}{x + 6})^{2} d x

\int ln (4 x + 1) d x

What is the integral of (x(x-1))/(sqrt(1-x)\sqrt{2-x)} ?
The integral of (x(x-1))/(sqrt(1-x)\sqrt{2-x)} is