What is the integral of (7x+5)/((x-2)(1-x)) ?

The integral of (7x+5)/((x-2)(1-x)) is -19ln|x-2|+12ln|x-1|+C

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integral of (7x+5)/((x-2)(1-x))

\int \frac{7 x + 5}{( x - 2 ) ( 1 - x )} d x

- 19 ln ∣ x - 2 ∣ + 12 ln ∣ x - 1 ∣ + C

Solution steps

\int \frac{7 x + 5}{( x - 2 ) ( 1 - x )} d x

Take the partial fraction of

\frac{7 x + 5}{( x - 2 ) ( 1 - x )} : - \frac{19}{x - 2} + \frac{12}{x - 1}

= \int - \frac{19}{x - 2} + \frac{12}{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{19}{x - 2} d x + \int \frac{12}{x - 1} d x

\int \frac{19}{x - 2} d x = 19 ln ∣ x - 2 ∣

\int \frac{12}{x - 1} d x = 12 ln ∣ x - 1 ∣

= - 19 ln ∣ x - 2 ∣ + 12 ln ∣ x - 1 ∣

Add a constant to the solution

= - 19 ln ∣ x - 2 ∣ + 12 ln ∣ x - 1 ∣ + C

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What is the integral of (7x+5)/((x-2)(1-x)) ?
The integral of (7x+5)/((x-2)(1-x)) is -19ln|x-2|+12ln|x-1|+C