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

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

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

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

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

Solution steps

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

Expand

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

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

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

\int 2 d x = 2 x

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

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

Add a constant to the solution

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

\frac{d}{d x} (20 x (1 - x)^{3})

\int \frac{1}{( 2 x + 5 ) ^{3}} d x

\int sin (2 x) x d x

9 x \frac{d y}{d x} = 2 x e^{x} - 9 y + 6 x^{2}

\frac{d}{d x} (\frac{2 π - x \cdot cos ( x )}{x - 2 π})

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