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

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

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

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

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

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

= 2 \cdot \int u^{2} - 2 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^{2} d u - \int 2 d u)

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

\int 2 d u = 2 u

= 2 (\frac{u ^{3}}{3} - 2 u)

Substitute back

u = 2 + x

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

Simplify

2 (\frac{( 2 + x ) ^{3}}{3} - 2 2 + x) : 2 (\frac{1}{3} (2 + x)^{\frac{3}{2}} - 2 2 + x)

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

Add a constant to the solution

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

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