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

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

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

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

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

Solution steps

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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