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

The integral of 5/((x-3)(2x+5)) is 5(1/11 ln|x-3|-1/11 ln|2x+5|)+C

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

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

5 (\frac{1}{11} ln ∣ x - 3 ∣ - \frac{1}{11} ln ∣ 2 x + 5 ∣) + C

Solution steps

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

Take the constant out:

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

= 5 \cdot \int \frac{1}{( x - 3 ) ( 2 x + 5 )} d x

Take the partial fraction of

\frac{1}{( x - 3 ) ( 2 x + 5 )} : \frac{1}{11 ( x - 3 )} - \frac{2}{11 ( 2 x + 5 )}

= 5 \cdot \int \frac{1}{11 ( x - 3 )} - \frac{2}{11 ( 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

= 5 (\int \frac{1}{11 ( x - 3 )} d x - \int \frac{2}{11 ( 2 x + 5 )} d x)

\int \frac{1}{11 ( x - 3 )} d x = \frac{1}{11} ln ∣ x - 3 ∣

\int \frac{2}{11 ( 2 x + 5 )} d x = \frac{1}{11} ln ∣ 2 x + 5 ∣

= 5 (\frac{1}{11} ln ∣ x - 3 ∣ - \frac{1}{11} ln ∣ 2 x + 5 ∣)

Add a constant to the solution

= 5 (\frac{1}{11} ln ∣ x - 3 ∣ - \frac{1}{11} ln ∣ 2 x + 5 ∣) + C

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What is the integral of 5/((x-3)(2x+5)) ?
The integral of 5/((x-3)(2x+5)) is 5(1/11 ln|x-3|-1/11 ln|2x+5|)+C