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

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

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

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

\frac{2}{3} x^{\frac{3}{2}} + \frac{4}{5} x^{\frac{5}{2}} + C

Solution steps

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

Expand

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

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

\int \frac{2 x ^{2}}{x} d x = \frac{4}{5} x^{\frac{5}{2}}

= \frac{2}{3} x^{\frac{3}{2}} + \frac{4}{5} x^{\frac{5}{2}}

Add a constant to the solution

= \frac{2}{3} x^{\frac{3}{2}} + \frac{4}{5} x^{\frac{5}{2}} + C

\int_{0}^{1} t cos (n (\frac{π}{2}) t) d t

x \to 3 lim (\frac{x + 3}{x - 3})

x \to 0 + lim (x^{18} ln (x))

x \to 3 - lim (\frac{1}{x - 3})

f (x) = \frac{5}{3 x}

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