What is the integral of (x^2)/(-x^2+4x-4) ?

The integral of (x^2)/(-x^2+4x-4) is -x-4ln|x-2|+4/(x-2)+C

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integral of (x^2)/(-x^2+4x-4)

\int \frac{x ^{2}}{- x ^{2} + 4 x - 4} d x

- x - 4 ln ∣ x - 2 ∣ + \frac{4}{x - 2} + C

Solution steps

\int \frac{x ^{2}}{- x ^{2} + 4 x - 4} d x

Take the partial fraction of

\frac{x ^{2}}{- x ^{2} + 4 x - 4} : - 1 - \frac{4}{x - 2} - \frac{4}{( x - 2 ) ^{2}}

= \int - 1 - \frac{4}{x - 2} - \frac{4}{( x - 2 ) ^{2}} 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 1 d x - \int \frac{4}{x - 2} d x - \int \frac{4}{( x - 2 ) ^{2}} d x

\int 1 d x = x

\int \frac{4}{x - 2} d x = 4 ln ∣ x - 2 ∣

\int \frac{4}{( x - 2 ) ^{2}} d x = - \frac{4}{x - 2}

= - x - 4 ln ∣ x - 2 ∣ - (- \frac{4}{x - 2})

Simplify

= - x - 4 ln ∣ x - 2 ∣ + \frac{4}{x - 2}

Add a constant to the solution

= - x - 4 ln ∣ x - 2 ∣ + \frac{4}{x - 2} + C

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What is the integral of (x^2)/(-x^2+4x-4) ?
The integral of (x^2)/(-x^2+4x-4) is -x-4ln|x-2|+4/(x-2)+C