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

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

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

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

\frac{2}{5} x^{\frac{5}{2}} + 4 x + C

Solution steps

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

Expand

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

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

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

\int \frac{2}{x} d x = 4 x

= \frac{2}{5} x^{\frac{5}{2}} + 4 x

Add a constant to the solution

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

x \to 2 lim (x + 5)

\int_{0}^{1} \frac{1}{x ^{\frac{18}{19}}} d x

x \to \frac{π}{4} lim (\frac{1 - 2 sin ( 2 x )}{cos ( x ) - sin ( x )})

\frac{d}{d x} (3 x^{2} - 5 x + 7)

\int cos (1 + x) d x

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