What is the integral of (4x+2)/(x+2) ?

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

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integral of (4x+2)/(x+2)

\int \frac{4 x + 2}{x + 2} d x

4 (x + 2) - 6 ln ∣ x + 2 ∣ + C

Solution steps

\int \frac{4 x + 2}{x + 2} d x

Apply u-substitution

: \int \frac{4 u - 6}{u} d u

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

Expand

\frac{4 u - 6}{u} : 4 - \frac{6}{u}

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

\int 4 d u = 4 u

\int \frac{6}{u} d u = 6 ln ∣ u ∣

= 4 u - 6 ln ∣ u ∣

Substitute back

u = x + 2

= 4 (x + 2) - 6 ln ∣ x + 2 ∣

Add a constant to the solution

= 4 (x + 2) - 6 ln ∣ x + 2 ∣ + C

\frac{\partial}{\partial y} (- \frac{x}{y ^{2}})

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

\int \frac{1}{( x - 5 ) ln ( x - 5 )} d x

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