What is the integral of x(x+4)^{-2} ?

The integral of x(x+4)^{-2} is ln|x+4|+4/(x+4)+C

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integral of x(x+4)^{-2}

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

ln ∣ x + 4 ∣ + \frac{4}{x + 4} + C

Solution steps

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

Simplify

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

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

Apply u-substitution

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

Expand

\frac{u - 4}{u ^{2}} : \frac{1}{u} - \frac{4}{u ^{2}}

= \int \frac{1}{u} - \frac{4}{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{4}{u ^{2}} d u

\int \frac{1}{u} d u = ln ∣ u ∣

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

= ln ∣ u ∣ - (- \frac{4}{u})

Substitute back

u = x + 4

= ln ∣ x + 4 ∣ - (- \frac{4}{x + 4})

Simplify

= ln ∣ x + 4 ∣ + \frac{4}{x + 4}

Add a constant to the solution

= ln ∣ x + 4 ∣ + \frac{4}{x + 4} + C

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