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

The integral of (2x)/((x+4)(x-5)) is 2(4/9 ln|x+4|+5/9 ln|x-5|)+C

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

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

2 (\frac{4}{9} ln ∣ x + 4 ∣ + \frac{5}{9} ln ∣ x - 5 ∣) + C

Solution steps

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

Take the constant out:

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

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

Take the partial fraction of

\frac{x}{( x + 4 ) ( x - 5 )} : \frac{4}{9 ( x + 4 )} + \frac{5}{9 ( x - 5 )}

= 2 \cdot \int \frac{4}{9 ( x + 4 )} + \frac{5}{9 ( x - 5 )} 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

= 2 (\int \frac{4}{9 ( x + 4 )} d x + \int \frac{5}{9 ( x - 5 )} d x)

\int \frac{4}{9 ( x + 4 )} d x = \frac{4}{9} ln ∣ x + 4 ∣

\int \frac{5}{9 ( x - 5 )} d x = \frac{5}{9} ln ∣ x - 5 ∣

= 2 (\frac{4}{9} ln ∣ x + 4 ∣ + \frac{5}{9} ln ∣ x - 5 ∣)

Add a constant to the solution

= 2 (\frac{4}{9} ln ∣ x + 4 ∣ + \frac{5}{9} ln ∣ x - 5 ∣) + C

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What is the integral of (2x)/((x+4)(x-5)) ?
The integral of (2x)/((x+4)(x-5)) is 2(4/9 ln|x+4|+5/9 ln|x-5|)+C