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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

Add a constant to the solution

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

x \to 2 lim (x^{2} - 2 x + 4)

\int cos (y x) d x

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

y = e^{2 x} + e^{- 2 x}

\frac{d y}{d x} = x + y, y (0) = 0

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