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

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

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

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

ln ∣ x + 2 ∣ + \frac{2}{x + 2} + C

Solution steps

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

Apply u-substitution

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

= \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 = x + 2

= ln ∣ x + 2 ∣ - (- \frac{2}{x + 2})

Simplify

= ln ∣ x + 2 ∣ + \frac{2}{x + 2}

Add a constant to the solution

= ln ∣ x + 2 ∣ + \frac{2}{x + 2} + C

3 x, x^{2}

\frac{d}{d x} (- 4 x^{\frac{4}{3}} - 2 x^{- \frac{3}{4}} + 3)

y = sec (x)

\int - e^{1 - x} d x

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

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