What is the integral of (5+x+x^2)/(sqrt(x)) ?

The integral of (5+x+x^2)/(sqrt(x)) is 10sqrt(x)+2/3 x^{3/2}+2/5 x^{5/2}+C

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integral of (5+x+x^2)/(sqrt(x))

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

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

Solution steps

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

Expand

\frac{5 + x + x ^{2}}{x} : \frac{5}{x} + x + x^{\frac{3}{2}}

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

\int \frac{5}{x} d x = 10 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}}

= 10 x + \frac{2}{3} x^{\frac{3}{2}} + \frac{2}{5} x^{\frac{5}{2}}

Add a constant to the solution

= 10 x + \frac{2}{3} x^{\frac{3}{2}} + \frac{2}{5} x^{\frac{5}{2}} + C

\int \frac{2 x + 11}{x ^{2} + 4 x + 5} d x

\frac{\partial}{\partial x} (8 x^{2} y - 3)

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

\int_{0}^{1} (2 x) d x

\frac{4 x - 9}{x - 9}

What is the integral of (5+x+x^2)/(sqrt(x)) ?
The integral of (5+x+x^2)/(sqrt(x)) is 10sqrt(x)+2/3 x^{3/2}+2/5 x^{5/2}+C