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

The integral of x/((2+x)^2) is ln|2+x|+2/(2+x)+C

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

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

ln ∣ 2 + x ∣ + \frac{2}{2 + x} + C

Solution steps

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

Apply u-substitution

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

Expand

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

= \int \frac{1}{u} - \frac{2}{u ^{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

= \int \frac{1}{u} d u - \int \frac{2}{u ^{2}} d u

\int \frac{1}{u} d u = ln ∣ u ∣

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

= ln ∣ u ∣ - (- \frac{2}{u})

Substitute back

u = 2 + x

= ln ∣ 2 + x ∣ - (- \frac{2}{2 + x})

Simplify

= ln ∣ 2 + x ∣ + \frac{2}{2 + x}

Add a constant to the solution

= ln ∣ 2 + x ∣ + \frac{2}{2 + x} + C

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