What is the integral of ((4+x+x^2))/(sqrt(x)) ?

The integral of ((4+x+x^2))/(sqrt(x)) is 8sqrt(x)+2/3 x^{3/2}+2/5 x^{5/2}+C

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integral of ((4+x+x^2))/(sqrt(x))

\int \frac{( 4 + x + x ^{2} )}{x} d x

8 x + \frac{2}{3} x^{\frac{3}{2}} + \frac{2}{5} x^{\frac{5}{2}} + C

Solution steps

\int \frac{4 + x + x ^{2}}{x} d x

Expand

\frac{4 + x + x ^{2}}{x} : \frac{4}{x} + x + x^{\frac{3}{2}}

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

\int \frac{4}{x} d x = 8 x

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

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

= 8 x + \frac{2}{3} x^{\frac{3}{2}} + \frac{2}{5} x^{\frac{5}{2}}

Add a constant to the solution

= 8 x + \frac{2}{3} x^{\frac{3}{2}} + \frac{2}{5} x^{\frac{5}{2}} + C

f (x) = ln (\frac{x ^{2} - 1}{x ^{2} + 1})

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

x^{3} y^{3} - y = x

\frac{d}{d x} (sin (x) + cos (y))

f (x) = \frac{3}{2 x - 4}

What is the integral of ((4+x+x^2))/(sqrt(x)) ?
The integral of ((4+x+x^2))/(sqrt(x)) is 8sqrt(x)+2/3 x^{3/2}+2/5 x^{5/2}+C