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

The integral of ((x^2+2))/((x+2)) is ((x+2)^2)/2-4(x+2)+6ln|x+2|+C

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

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

\frac{( x + 2 ) ^{2}}{2} - 4 (x + 2) + 6 ln ∣ x + 2 ∣ + C

Solution steps

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

Apply u-substitution

: \int \frac{u ^{2} - 4 u + 6}{u} d u

= \int \frac{u ^{2} - 4 u + 6}{u} d u

Expand

\frac{u ^{2} - 4 u + 6}{u} : u - 4 + \frac{6}{u}

= \int u - 4 + \frac{6}{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 u d u - \int 4 d u + \int \frac{6}{u} d u

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

\int 4 d u = 4 u

\int \frac{6}{u} d u = 6 ln ∣ u ∣

= \frac{u ^{2}}{2} - 4 u + 6 ln ∣ u ∣

Substitute back

u = x + 2

= \frac{( x + 2 ) ^{2}}{2} - 4 (x + 2) + 6 ln ∣ x + 2 ∣

Add a constant to the solution

= \frac{( x + 2 ) ^{2}}{2} - 4 (x + 2) + 6 ln ∣ x + 2 ∣ + C

x \to 0 lim (6 x + ∣ x - 5 ∣)

\frac{d}{d x} (\frac{2 x ^{2} + 2 x + 8}{x})

f (x) = lo g_{2} (x)

x \to - 3 - lim (f (x))

\int 2 x + sin (x) + c d x

What is the integral of ((x^2+2))/((x+2)) ?
The integral of ((x^2+2))/((x+2)) is ((x+2)^2)/2-4(x+2)+6ln|x+2|+C