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

The integral of (sqrt(x))/(2+sqrt(x)) is x-4sqrt(x)+8ln|sqrt(x)+2|-12+C

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

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

x - 4 x + 8 ln x + 2 - 12 + C

Solution steps

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

Apply u-substitution

: \int \frac{2 ( u ^{2} - 4 u + 4 )}{u} d u

= \int \frac{2 ( u ^{2} - 4 u + 4 )}{u} d u

Take the constant out:

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

= 2 \cdot \int \frac{u ^{2} - 4 u + 4}{u} d u

Expand

\frac{u ^{2} - 4 u + 4}{u} : u - 4 + \frac{4}{u}

= 2 \cdot \int u - 4 + \frac{4}{u} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int u d u - \int 4 d u + \int \frac{4}{u} d u)

\int u d u = \frac{u ^{2}}{2}

\int 4 d u = 4 u

\int \frac{4}{u} d u = 4 ln ∣ u ∣

= 2 (\frac{u ^{2}}{2} - 4 u + 4 ln ∣ u ∣)

Substitute back

u = 2 + x

= 2 (\frac{( 2 + x ) ^{2}}{2} - 4 (2 + x) + 4 ln 2 + x)

Simplify

2 (\frac{( 2 + x ) ^{2}}{2} - 4 (2 + x) + 4 ln 2 + x) : x - 4 x + 8 ln x + 2 - 12

= x - 4 x + 8 ln x + 2 - 12

Add a constant to the solution

= x - 4 x + 8 ln x + 2 - 12 + C

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What is the integral of (sqrt(x))/(2+sqrt(x)) ?
The integral of (sqrt(x))/(2+sqrt(x)) is x-4sqrt(x)+8ln|sqrt(x)+2|-12+C