What is the integral of (x-3)/(sqrt(x)) ?

The integral of (x-3)/(sqrt(x)) is 2/3 x^{3/2}-6sqrt(x)+C

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integral of (x-3)/(sqrt(x))

\int \frac{x - 3}{x} d x

\frac{2}{3} x^{\frac{3}{2}} - 6 x + C

Solution steps

\int \frac{x - 3}{x} d x

Expand

\frac{x - 3}{x} : x - \frac{3}{x}

= \int x - \frac{3}{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 \frac{3}{x} d x

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

\int \frac{3}{x} d x = 6 x

= \frac{2}{3} x^{\frac{3}{2}} - 6 x

Add a constant to the solution

= \frac{2}{3} x^{\frac{3}{2}} - 6 x + C

\frac{d}{d x} (2 x^{2} + 8 x)

x (x^{2} - 11 x + 35), x^{2}

\frac{d}{d x} (\frac{3}{4} x^{12})

y^{^{'}} = - \frac{1}{x} y^{2} - \frac{1}{x} y

x^{\frac{2}{3}} + y^{\frac{2}{3}} = 5, (- 8, - 1)

What is the integral of (x-3)/(sqrt(x)) ?
The integral of (x-3)/(sqrt(x)) is 2/3 x^{3/2}-6sqrt(x)+C