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

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

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

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

4 (2 + x - 2 ln 2 + x) + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

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

Take the constant out:

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

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

Expand

\frac{u - 2}{u} : 1 - \frac{2}{u}

= 2 \cdot 2 \cdot \int 1 - \frac{2}{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 \cdot 2 (\int 1 d u - \int \frac{2}{u} d u)

\int 1 d u = u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

= 2 \cdot 2 (u - 2 ln ∣ u ∣)

Substitute back

u = 2 + x

= 2 \cdot 2 (2 + x - 2 ln 2 + x)

Simplify

= 4 (2 + x - 2 ln 2 + x)

Add a constant to the solution

= 4 (2 + x - 2 ln 2 + x) + C

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