What is the integral of ((2x+3))/(x+2) ?

The integral of ((2x+3))/(x+2) is 2(x+2)-ln|x+2|+C

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integral of ((2x+3))/(x+2)

\int \frac{( 2 x + 3 )}{x + 2} d x

2 (x + 2) - ln ∣ x + 2 ∣ + C

Solution steps

\int \frac{2 x + 3}{x + 2} d x

Apply u-substitution

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

Expand

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

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

\int 2 d u = 2 u

\int \frac{1}{u} d u = ln ∣ u ∣

= 2 u - ln ∣ u ∣

Substitute back

u = x + 2

= 2 (x + 2) - ln ∣ x + 2 ∣

Add a constant to the solution

= 2 (x + 2) - ln ∣ x + 2 ∣ + C

What is the integral of ((2x+3))/(x+2) ?
The integral of ((2x+3))/(x+2) is 2(x+2)-ln|x+2|+C