What is the integral of (82x+8)/((9x+1)(x-1)) ?

The integral of (82x+8)/((9x+1)(x-1)) is 1/9 ln|9x+1|+9ln|x-1|+C

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integral of (82x+8)/((9x+1)(x-1))

\int \frac{82 x + 8}{( 9 x + 1 ) ( x - 1 )} d x

\frac{1}{9} ln ∣ 9 x + 1 ∣ + 9 ln ∣ x - 1 ∣ + C

Solution steps

\int \frac{82 x + 8}{( 9 x + 1 ) ( x - 1 )} d x

Take the partial fraction of

\frac{82 x + 8}{( 9 x + 1 ) ( x - 1 )} : \frac{1}{9 x + 1} + \frac{9}{x - 1}

= \int \frac{1}{9 x + 1} + \frac{9}{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{1}{9 x + 1} d x + \int \frac{9}{x - 1} d x

\int \frac{1}{9 x + 1} d x = \frac{1}{9} ln ∣ 9 x + 1 ∣

\int \frac{9}{x - 1} d x = 9 ln ∣ x - 1 ∣

= \frac{1}{9} ln ∣ 9 x + 1 ∣ + 9 ln ∣ x - 1 ∣

Add a constant to the solution

= \frac{1}{9} ln ∣ 9 x + 1 ∣ + 9 ln ∣ x - 1 ∣ + C

What is the integral of (82x+8)/((9x+1)(x-1)) ?
The integral of (82x+8)/((9x+1)(x-1)) is 1/9 ln|9x+1|+9ln|x-1|+C