What is the integral of 1/((2x-1)(x+5)) ?

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

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integral of 1/((2x-1)(x+5))

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

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

Solution steps

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

Take the partial fraction of

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

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

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

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

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

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

Add a constant to the solution

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

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