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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Expand

\frac{1 - x}{x} : \frac{1}{x} - x

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

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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