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

\frac{\partial}{\partial x} (9)

y^{^{'}} = a^{x + y}

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

\frac{d y}{d x} = x^{2} y^{- 5}

(2 x y^{2} + 3 x^{2}) d x + (2 x^{2} y) d y = 0

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