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

The integral of (x-3)/(x+2) is x+2-5ln|x+2|+C

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

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

x + 2 - 5 ln ∣ x + 2 ∣ + C

Solution steps

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

Apply u-substitution

: \int \frac{u - 5}{u} d u

= \int \frac{u - 5}{u} d u

Expand

\frac{u - 5}{u} : 1 - \frac{5}{u}

= \int 1 - \frac{5}{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 1 d u - \int \frac{5}{u} d u

\int 1 d u = u

\int \frac{5}{u} d u = 5 ln ∣ u ∣

= u - 5 ln ∣ u ∣

Substitute back

u = x + 2

= x + 2 - 5 ln ∣ x + 2 ∣

Add a constant to the solution

= x + 2 - 5 ln ∣ x + 2 ∣ + C

\int ln (x^{2} + 1) x d x

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

What is the integral of (x-3)/(x+2) ?
The integral of (x-3)/(x+2) is x+2-5ln|x+2|+C