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

The integral of (x+5)/((x+2)(x^2-2x+2)^2) is

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integral of (x+5)/((x+2)(x^2-2x+2)^2)

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

\frac{3}{100} ln ∣ x + 2 ∣ + \frac{1}{200} (- 3 ln x^{2} - 2 x + 2 + 18 arctan (x - 1)) + \frac{1}{10} (- 3 (\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 )}) + 11 (arctan (x - 1) + \frac{1}{2} sin (2 arctan (x - 1)))) + C

Solution steps

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

Take the partial fraction of

\frac{x + 5}{( x + 2 ) ( x ^{2} - 2 x + 2 ) ^{2}} : \frac{3}{100 ( x + 2 )} + \frac{- 3 x + 12}{100 ( x ^{2} - 2 x + 2 )} + \frac{- 3 x + 22}{10 ( x ^{2} - 2 x + 2 ) ^{2}}

= \int \frac{3}{100 ( x + 2 )} + \frac{- 3 x + 12}{100 ( x ^{2} - 2 x + 2 )} + \frac{- 3 x + 22}{10 ( x ^{2} - 2 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 \frac{3}{100 ( x + 2 )} d x + \int \frac{- 3 x + 12}{100 ( x ^{2} - 2 x + 2 )} d x + \int \frac{- 3 x + 22}{10 ( x ^{2} - 2 x + 2 ) ^{2}} d x

\int \frac{3}{100 ( x + 2 )} d x = \frac{3}{100} ln ∣ x + 2 ∣

\int \frac{- 3 x + 12}{100 ( x ^{2} - 2 x + 2 )} d x = \frac{1}{200} (- 3 ln x^{2} - 2 x + 2 + 18 arctan (x - 1))

\int \frac{- 3 x + 22}{10 ( x ^{2} - 2 x + 2 ) ^{2}} d x = \frac{1}{10} (- 3 (\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 )}) + 11 (arctan (x - 1) + \frac{1}{2} sin (2 arctan (x - 1))))

= \frac{3}{100} ln ∣ x + 2 ∣ + \frac{1}{200} (- 3 ln x^{2} - 2 x + 2 + 18 arctan (x - 1)) + \frac{1}{10} (- 3 (\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 )}) + 11 (arctan (x - 1) + \frac{1}{2} sin (2 arctan (x - 1))))

Add a constant to the solution

= \frac{3}{100} ln ∣ x + 2 ∣ + \frac{1}{200} (- 3 ln x^{2} - 2 x + 2 + 18 arctan (x - 1)) + \frac{1}{10} (- 3 (\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 )}) + 11 (arctan (x - 1) + \frac{1}{2} sin (2 arctan (x - 1)))) + C

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What is the integral of (x+5)/((x+2)(x^2-2x+2)^2) ?
The integral of (x+5)/((x+2)(x^2-2x+2)^2) is