What is the integral of ((2x+3))/((x-2)(x+5)) ?

The integral of ((2x+3))/((x-2)(x+5)) is ln(|x-2||x+5|)+C

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integral of ((2x+3))/((x-2)(x+5))

\int \frac{( 2 x + 3 )}{( x - 2 ) ( x + 5 )} d x

ln (∣ x - 2 ∣ ∣ x + 5 ∣) + C

Solution steps

\int \frac{2 x + 3}{( x - 2 ) ( x + 5 )} d x

Apply u-substitution

: \int \frac{1}{u} d u

= \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = (x - 2) (x + 5)

= ln ∣ (x - 2) (x + 5) ∣

Simplify

ln ∣ (x - 2) (x + 5) ∣ : ln (∣ x - 2 ∣ ∣ x + 5 ∣)

= ln (∣ x - 2 ∣ ∣ x + 5 ∣)

Add a constant to the solution

= ln (∣ x - 2 ∣ ∣ x + 5 ∣) + C

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What is the integral of ((2x+3))/((x-2)(x+5)) ?
The integral of ((2x+3))/((x-2)(x+5)) is ln(|x-2||x+5|)+C