What is the integral of (x+2)/(x(x^2-4x+5)^2) ?

The integral of (x+2)/(x(x^2-4x+5)^2) is

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integral of (x+2)/(x(x^2-4x+5)^2)

\int \frac{x + 2}{x ( x ^{2} - 4 x + 5 ) ^{2}} d x

\frac{2}{25} ln ∣ x ∣ - \frac{1}{25} ln x^{2} - 4 x + 5 + \frac{53}{50} arctan (x - 2) + \frac{13}{20} sin (2 arctan (x - 2)) + \frac{1}{5 ( x ^{2} - 4 x + 5 )} - \frac{2 ( x - 2 )}{5 ( x ^{2} - 4 x + 5 )} + C

Solution steps

\int \frac{x + 2}{x ( x ^{2} - 4 x + 5 ) ^{2}} d x

Take the partial fraction of

\frac{x + 2}{x ( x ^{2} - 4 x + 5 ) ^{2}} : \frac{2}{25 x} + \frac{- 2 x + 8}{25 ( x ^{2} - 4 x + 5 )} + \frac{- 2 x + 13}{5 ( x ^{2} - 4 x + 5 ) ^{2}}

= \int \frac{2}{25 x} + \frac{- 2 x + 8}{25 ( x ^{2} - 4 x + 5 )} + \frac{- 2 x + 13}{5 ( x ^{2} - 4 x + 5 ) ^{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{2}{25 x} d x + \int \frac{- 2 x + 8}{25 ( x ^{2} - 4 x + 5 )} d x + \int \frac{- 2 x + 13}{5 ( x ^{2} - 4 x + 5 ) ^{2}} d x

\int \frac{2}{25 x} d x = \frac{2}{25} ln ∣ x ∣

\int \frac{- 2 x + 8}{25 ( x ^{2} - 4 x + 5 )} d x = \frac{1}{25} (- ln x^{2} - 4 x + 5 + 4 arctan (x - 2))

\int \frac{- 2 x + 13}{5 ( x ^{2} - 4 x + 5 ) ^{2}} d x = \frac{1}{5} (\frac{9}{2} arctan (x - 2) + \frac{13}{4} sin (2 arctan (x - 2)) + \frac{1}{x ^{2} - 4 x + 5} - \frac{2 ( x - 2 )}{x ^{2} - 4 x + 5})

= \frac{2}{25} ln ∣ x ∣ + \frac{1}{25} (- ln x^{2} - 4 x + 5 + 4 arctan (x - 2)) + \frac{1}{5} (\frac{9}{2} arctan (x - 2) + \frac{13}{4} sin (2 arctan (x - 2)) + \frac{1}{x ^{2} - 4 x + 5} - \frac{2 ( x - 2 )}{x ^{2} - 4 x + 5})

Simplify

\frac{2}{25} ln ∣ x ∣ + \frac{1}{25} (- ln x^{2} - 4 x + 5 + 4 arctan (x - 2)) + \frac{1}{5} (\frac{9}{2} arctan (x - 2) + \frac{13}{4} sin (2 arctan (x - 2)) + \frac{1}{x ^{2} - 4 x + 5} - \frac{2 ( x - 2 )}{x ^{2} - 4 x + 5}) : \frac{2}{25} ln ∣ x ∣ - \frac{1}{25} ln x^{2} - 4 x + 5 + \frac{53}{50} arctan (x - 2) + \frac{13}{20} sin (2 arctan (x - 2)) + \frac{1}{5 ( x ^{2} - 4 x + 5 )} - \frac{2 ( x - 2 )}{5 ( x ^{2} - 4 x + 5 )}

= \frac{2}{25} ln ∣ x ∣ - \frac{1}{25} ln x^{2} - 4 x + 5 + \frac{53}{50} arctan (x - 2) + \frac{13}{20} sin (2 arctan (x - 2)) + \frac{1}{5 ( x ^{2} - 4 x + 5 )} - \frac{2 ( x - 2 )}{5 ( x ^{2} - 4 x + 5 )}

Add a constant to the solution

= \frac{2}{25} ln ∣ x ∣ - \frac{1}{25} ln x^{2} - 4 x + 5 + \frac{53}{50} arctan (x - 2) + \frac{13}{20} sin (2 arctan (x - 2)) + \frac{1}{5 ( x ^{2} - 4 x + 5 )} - \frac{2 ( x - 2 )}{5 ( x ^{2} - 4 x + 5 )} + C

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What is the integral of (x+2)/(x(x^2-4x+5)^2) ?
The integral of (x+2)/(x(x^2-4x+5)^2) is