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

The integral of ((sqrt(x)+2)^2)/(2sqrt(x)) is 1/3 (x^{3/2}+6x+12sqrt(x))+C

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

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

\frac{1}{3} (x^{\frac{3}{2}} + 6 x + 12 x) + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Expand

\frac{( x + 2 ) ^{2}}{x} : x + 4 + \frac{4}{x}

= \frac{1}{2} \cdot \int x + 4 + \frac{4}{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

= \frac{1}{2} (\int x d x + \int 4 d x + \int \frac{4}{x} d x)

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

\int 4 d x = 4 x

\int \frac{4}{x} d x = 8 x

= \frac{1}{2} (\frac{2}{3} x^{\frac{3}{2}} + 4 x + 8 x)

Simplify

\frac{1}{2} (\frac{2}{3} x^{\frac{3}{2}} + 4 x + 8 x) : \frac{1}{3} (x^{\frac{3}{2}} + 6 x + 12 x)

= \frac{1}{3} (x^{\frac{3}{2}} + 6 x + 12 x)

Add a constant to the solution

= \frac{1}{3} (x^{\frac{3}{2}} + 6 x + 12 x) + C

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\int \frac{6}{2 x + 1} d x

What is the integral of ((sqrt(x)+2)^2)/(2sqrt(x)) ?
The integral of ((sqrt(x)+2)^2)/(2sqrt(x)) is 1/3 (x^{3/2}+6x+12sqrt(x))+C