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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Add a constant to the solution

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

\int lo g_{4} (3 x + 1) d x

3 y^{^{''}} + 2 y^{^{'}} - y = 0

\int x (2 x + 1) e^{4 x^{3} + 3 x^{2} - 4} d x

y^{^{'}} + y^{2} = 1

\frac{( 4 x ^{5} + 5 )}{x ^{3}}

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