What is the integral of (x^2-3x+5)/(sqrt(x)) ?

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

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integral of (x^2-3x+5)/(sqrt(x))

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

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

Solution steps

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

Expand

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

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

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

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

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

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

Add a constant to the solution

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

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\int \frac{( arctan ( x ) ) ^{3}}{x ^{2} + 1} d x

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