What is the integral of 2/((x-7)(x-5)) ?

The integral of 2/((x-7)(x-5)) is ln|x-7|-ln|x-5|+C

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integral of 2/((x-7)(x-5))

\int \frac{2}{( x - 7 ) ( x - 5 )} d x

ln ∣ x - 7 ∣ - ln ∣ x - 5 ∣ + C

Solution steps

\int \frac{2}{( x - 7 ) ( 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{1}{( x - 7 ) ( x - 5 )} d x

Take the partial fraction of

\frac{1}{( x - 7 ) ( x - 5 )} : \frac{1}{2 ( x - 7 )} - \frac{1}{2 ( x - 5 )}

= 2 \cdot \int \frac{1}{2 ( x - 7 )} - \frac{1}{2 ( 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{1}{2 ( x - 7 )} d x - \int \frac{1}{2 ( x - 5 )} d x)

\int \frac{1}{2 ( x - 7 )} d x = \frac{1}{2} ln ∣ x - 7 ∣

\int \frac{1}{2 ( x - 5 )} d x = \frac{1}{2} ln ∣ x - 5 ∣

= 2 (\frac{1}{2} ln ∣ x - 7 ∣ - \frac{1}{2} ln ∣ x - 5 ∣)

Simplify

2 (\frac{1}{2} ln ∣ x - 7 ∣ - \frac{1}{2} ln ∣ x - 5 ∣) : ln ∣ x - 7 ∣ - ln ∣ x - 5 ∣

= ln ∣ x - 7 ∣ - ln ∣ x - 5 ∣

Add a constant to the solution

= ln ∣ x - 7 ∣ - ln ∣ x - 5 ∣ + C

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What is the integral of 2/((x-7)(x-5)) ?
The integral of 2/((x-7)(x-5)) is ln|x-7|-ln|x-5|+C