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

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

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integral of (x/(x+2))

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

x + 2 - 2 ln ∣ x + 2 ∣ + C

Solution steps

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

Apply u-substitution

: \int \frac{u - 2}{u} d u

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

Expand

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

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

\int 1 d u = u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

= u - 2 ln ∣ u ∣

Substitute back

u = x + 2

= x + 2 - 2 ln ∣ x + 2 ∣

Add a constant to the solution

= x + 2 - 2 ln ∣ x + 2 ∣ + C

x \to 7 lim (\frac{x - 8}{x - 7})

y^{^{''}} = y - 3 x

\frac{d}{d x} (\frac{5}{3} x^{3} - x^{2} + 1)

x \to - 1 lim (4 x^{3} - 2 x^{2} + 2)

f (x) = x^{2} - x + 4