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

The integral of ((1+sqrt(x))^2)/(sqrt(x)) is 2sqrt(x)+2x+2/3 x^{3/2}+C

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

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

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

Solution steps

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

Expand

\frac{( 1 + x ) ^{2}}{x} : \frac{1}{x} + 2 + x

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

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

\int 2 d x = 2 x

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

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

Add a constant to the solution

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

\int 1 + 4 x - x^{2} d x

\int arctan (\frac{1}{x}) d x

\frac{d}{d x} (2 x^{2} - 4 x + 3)

f (x) = 139 + \frac{36}{x} + \frac{x ^{2}}{16}

x^{^{''}} - 6 x^{^{'}} + 25 x = 64 e^{- t}

What is the integral of ((1+sqrt(x))^2)/(sqrt(x)) ?
The integral of ((1+sqrt(x))^2)/(sqrt(x)) is 2sqrt(x)+2x+2/3 x^{3/2}+C