What is the integral of (5x)/((x+4)(x+2)) ?

The integral of (5x)/((x+4)(x+2)) is 5(2ln|x+4|-ln|x+2|)+C

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integral of (5x)/((x+4)(x+2))

\int \frac{5 x}{( x + 4 ) ( x + 2 )} d x

5 (2 ln ∣ x + 4 ∣ - ln ∣ x + 2 ∣) + C

Solution steps

\int \frac{5 x}{( x + 4 ) ( x + 2 )} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \int \frac{x}{( x + 4 ) ( x + 2 )} d x

Take the partial fraction of

\frac{x}{( x + 4 ) ( x + 2 )} : \frac{2}{x + 4} - \frac{1}{x + 2}

= 5 \cdot \int \frac{2}{x + 4} - \frac{1}{x + 2} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 5 (\int \frac{2}{x + 4} d x - \int \frac{1}{x + 2} d x)

\int \frac{2}{x + 4} d x = 2 ln ∣ x + 4 ∣

\int \frac{1}{x + 2} d x = ln ∣ x + 2 ∣

= 5 (2 ln ∣ x + 4 ∣ - ln ∣ x + 2 ∣)

Add a constant to the solution

= 5 (2 ln ∣ x + 4 ∣ - ln ∣ x + 2 ∣) + C

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What is the integral of (5x)/((x+4)(x+2)) ?
The integral of (5x)/((x+4)(x+2)) is 5(2ln|x+4|-ln|x+2|)+C