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

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

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integral of (x^2)/((2x-x^2))

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

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

Solution steps

\int \frac{x ^{2}}{2 x - x ^{2}} d x

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

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