What is the integral of ((x^2+1)/(sqrt(x))) ?

The integral of ((x^2+1)/(sqrt(x))) is 2/5 x^{5/2}+2sqrt(x)+C

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integral of ((x^2+1)/(sqrt(x)))

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

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

Solution steps

\int \frac{x ^{2} + 1}{x} d x

Expand

\frac{x ^{2} + 1}{x} : x^{\frac{3}{2}} + \frac{1}{x}

= \int x^{\frac{3}{2}} + \frac{1}{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 \frac{1}{x} d x

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

\int \frac{1}{x} d x = 2 x

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

Add a constant to the solution

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

\int \frac{1}{3 x ^{2} + 5} d x

x^{2} y^{^{''}} - 7 x y^{^{'}} + 16 y = 0

y = - 3 x^{3} + x^{2} + 10 x, x \geq 0

y^{^{''}} - 2 y^{^{'}} - 3 y = x e^{x}

e^{\frac{4}{5} x}

What is the integral of ((x^2+1)/(sqrt(x))) ?
The integral of ((x^2+1)/(sqrt(x))) is 2/5 x^{5/2}+2sqrt(x)+C