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

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

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

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

\frac{2}{3} x^{\frac{3}{2}} + 2 (x + 2 - 2 ln x + 2) + C

Solution steps

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

Long division

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

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

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

\int \frac{1}{x + 2} d x = 2 (x + 2 - 2 ln x + 2)

= \frac{2}{3} x^{\frac{3}{2}} + 2 (x + 2 - 2 ln x + 2)

Add a constant to the solution

= \frac{2}{3} x^{\frac{3}{2}} + 2 (x + 2 - 2 ln x + 2) + C

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