What is the integral of (8+x^2)/(sqrt(x)) ?

The integral of (8+x^2)/(sqrt(x)) is 16sqrt(x)+2/5 x^{5/2}+C

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integral of (8+x^2)/(sqrt(x))

\int \frac{8 + x ^{2}}{x} d x

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

Solution steps

\int \frac{8 + x ^{2}}{x} d x

Expand

\frac{8 + x ^{2}}{x} : \frac{8}{x} + x^{\frac{3}{2}}

= \int \frac{8}{x} + x^{\frac{3}{2}} 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{8}{x} d x + \int x^{\frac{3}{2}} d x

\int \frac{8}{x} d x = 16 x

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

= 16 x + \frac{2}{5} x^{\frac{5}{2}}

Add a constant to the solution

= 16 x + \frac{2}{5} x^{\frac{5}{2}} + C

\int_{- 2}^{2} (x^{2} + 1) d x

2 y^{^{''}} + 7 y^{^{'}} + 3 y = 0

f (x) = x^{2} - 2 x^{3}

\frac{d}{d x} (2 x^{2} sin (x))

- 8 e^{- 4 x^{2}} x

What is the integral of (8+x^2)/(sqrt(x)) ?
The integral of (8+x^2)/(sqrt(x)) is 16sqrt(x)+2/5 x^{5/2}+C