What is the integral of ((2x^2-x+3))/(sqrt(x)) ?

The integral of ((2x^2-x+3))/(sqrt(x)) is 4/5 x^{5/2}-2/3 x^{3/2}+6sqrt(x)+C

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integral of ((2x^2-x+3))/(sqrt(x))

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

\frac{4}{5} x^{\frac{5}{2}} - \frac{2}{3} x^{\frac{3}{2}} + 6 x + C

Solution steps

\int \frac{2 x ^{2} - x + 3}{x} d x

Expand

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

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

\int 2 x^{\frac{3}{2}} d x = \frac{4}{5} x^{\frac{5}{2}}

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

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

= \frac{4}{5} x^{\frac{5}{2}} - \frac{2}{3} x^{\frac{3}{2}} + 6 x

Add a constant to the solution

= \frac{4}{5} x^{\frac{5}{2}} - \frac{2}{3} x^{\frac{3}{2}} + 6 x + C

s l o p e y = 2 x^{3} - x^{2} + 3

\frac{d y}{d x} = 4 x y

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

d er i v a t i v e 3 - 5 x

d er i v a t i v e f (x) = a x^{2} + b x

What is the integral of ((2x^2-x+3))/(sqrt(x)) ?
The integral of ((2x^2-x+3))/(sqrt(x)) is 4/5 x^{5/2}-2/3 x^{3/2}+6sqrt(x)+C