What is the integral of (5x-2)/((x+2)^2) ?

The integral of (5x-2)/((x+2)^2) is 5ln|x+2|+(12)/(x+2)+C

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integral of (5x-2)/((x+2)^2)

\int \frac{5 x - 2}{( x + 2 ) ^{2}} d x

5 ln ∣ x + 2 ∣ + \frac{12}{x + 2} + C

Solution steps

\int \frac{5 x - 2}{( x + 2 ) ^{2}} d x

Apply u-substitution

: \int \frac{5 u - 12}{u ^{2}} d u

= \int \frac{5 u - 12}{u ^{2}} d u

Expand

\frac{5 u - 12}{u ^{2}} : \frac{5}{u} - \frac{12}{u ^{2}}

= \int \frac{5}{u} - \frac{12}{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{5}{u} d u - \int \frac{12}{u ^{2}} d u

\int \frac{5}{u} d u = 5 ln ∣ u ∣

\int \frac{12}{u ^{2}} d u = - \frac{12}{u}

= 5 ln ∣ u ∣ - (- \frac{12}{u})

Substitute back

u = x + 2

= 5 ln ∣ x + 2 ∣ - (- \frac{12}{x + 2})

Simplify

= 5 ln ∣ x + 2 ∣ + \frac{12}{x + 2}

Add a constant to the solution

= 5 ln ∣ x + 2 ∣ + \frac{12}{x + 2} + C

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What is the integral of (5x-2)/((x+2)^2) ?
The integral of (5x-2)/((x+2)^2) is 5ln|x+2|+(12)/(x+2)+C