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

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

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

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

2 x - \frac{2}{3} x^{\frac{3}{2}} + C

Solution steps

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

Expand

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

= \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

= \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}}

= 2 x - \frac{2}{3} x^{\frac{3}{2}}

Add a constant to the solution

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

\frac{d}{d x} (3 x^{2} + \frac{2}{x})

\frac{d}{d x} (2 x - y)

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

x \to 2 + lim (\frac{x}{x - 2})

x \to 4 lim (3 x - 1)

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