What is the integral of (4(x-6))/((x^2-2x+2)^2) ?

The integral of (4(x-6))/((x^2-2x+2)^2) is

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integral of (4(x-6))/((x^2-2x+2)^2)

\int \frac{4 ( x - 6 )}{( x ^{2} - 2 x + 2 ) ^{2}} d x

4 (\frac{x ( arctan ( x - 1 ) ( x ^{2} - 2 x + 2 ) + x - 1 )}{2 ( x ^{2} - 2 x + 2 )} - \frac{( x - 1 ) ( arctan ( x - 1 ) + \frac{x - 1}{x ^{2} - 2 x + 2} ) ( x ^{2} - 2 x + 2 ) + 1}{2 ( x ^{2} - 2 x + 2 )} - 3 (arctan (x - 1) + \frac{1}{2} sin (2 arctan (x - 1)))) + C

Solution steps

\int \frac{4 ( x - 6 )}{( x ^{2} - 2 x + 2 ) ^{2}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 \cdot \int \frac{x - 6}{( x ^{2} - 2 x + 2 ) ^{2}} d x

Expand

\frac{x - 6}{( x ^{2} - 2 x + 2 ) ^{2}} : \frac{x}{( x ^{2} - 2 x + 2 ) ^{2}} - \frac{6}{( x ^{2} - 2 x + 2 ) ^{2}}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 4 (\int \frac{x}{( x ^{2} - 2 x + 2 ) ^{2}} d x - \int \frac{6}{( x ^{2} - 2 x + 2 ) ^{2}} d x)

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

\int \frac{6}{( x ^{2} - 2 x + 2 ) ^{2}} d x = 3 (arctan (x - 1) + \frac{1}{2} sin (2 arctan (x - 1)))

= 4 (\frac{x ( arctan ( x - 1 ) ( x ^{2} - 2 x + 2 ) + x - 1 )}{2 ( x ^{2} - 2 x + 2 )} - \frac{( x - 1 ) ( arctan ( x - 1 ) + \frac{x - 1}{x ^{2} - 2 x + 2} ) ( x ^{2} - 2 x + 2 ) + 1}{2 ( x ^{2} - 2 x + 2 )} - 3 (arctan (x - 1) + \frac{1}{2} sin (2 arctan (x - 1))))

Add a constant to the solution

= 4 (\frac{x ( arctan ( x - 1 ) ( x ^{2} - 2 x + 2 ) + x - 1 )}{2 ( x ^{2} - 2 x + 2 )} - \frac{( x - 1 ) ( arctan ( x - 1 ) + \frac{x - 1}{x ^{2} - 2 x + 2} ) ( x ^{2} - 2 x + 2 ) + 1}{2 ( x ^{2} - 2 x + 2 )} - 3 (arctan (x - 1) + \frac{1}{2} sin (2 arctan (x - 1)))) + C

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

n = 0 \sum \infty \frac{2 ^{n} z ^{n}}{n}

\frac{d}{d x} (x^{4} e^{8 x})

\frac{d}{d x} (I n (x^{2} + 1))

\frac{\partial}{\partial z} (2 x z^{2})

What is the integral of (4(x-6))/((x^2-2x+2)^2) ?
The integral of (4(x-6))/((x^2-2x+2)^2) is