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

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

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

\int \frac{x}{x - 3 x + 2} d x

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

Solution steps

\int \frac{x}{x - 3 x + 2} d x

Simplify

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

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

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