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

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

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

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

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

Solution steps

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

Take the partial fraction of

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

= \int \frac{2}{x - 1} + \frac{1}{2 x + 1} 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}{x - 1} d x + \int \frac{1}{2 x + 1} d x

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

\int \frac{1}{2 x + 1} d x = \frac{1}{2} ln ∣ 2 x + 1 ∣

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

Add a constant to the solution

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

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