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 - 4 arctan (x - 2)) + \frac{x ^{4} arctan ( x - 2 ) + x ^{3} - 8 x ^{3} arctan ( x - 2 ) + 26 x ^{2} arctan ( x - 2 ) - 8 x ^{2} + 21 x - 40 x arctan ( x - 2 ) + 25 arctan ( x - 2 ) - 20}{10 ( x ^{2} - 4 x + 5 ) ^{2}} + 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 - 3}{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 - 3}{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 - 3}{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 - 3}{5 ( x ^{2} - 4 x + 5 ) ^{2}} d x = \frac{x ^{4} arctan ( x - 2 ) + x ^{3} - 8 x ^{3} arctan ( x - 2 ) + 26 x ^{2} arctan ( x - 2 ) - 8 x ^{2} + 21 x - 40 x arctan ( x - 2 ) + 25 arctan ( x - 2 ) - 20}{10 ( x ^{2} - 4 x + 5 ) ^{2}}

= - \frac{2}{25} ln ∣ x ∣ + \frac{1}{25} (ln x^{2} - 4 x + 5 - 4 arctan (x - 2)) + \frac{x ^{4} arctan ( x - 2 ) + x ^{3} - 8 x ^{3} arctan ( x - 2 ) + 26 x ^{2} arctan ( x - 2 ) - 8 x ^{2} + 21 x - 40 x arctan ( x - 2 ) + 25 arctan ( x - 2 ) - 20}{10 ( x ^{2} - 4 x + 5 ) ^{2}}

Add a constant to the solution

= - \frac{2}{25} ln ∣ x ∣ + \frac{1}{25} (ln x^{2} - 4 x + 5 - 4 arctan (x - 2)) + \frac{x ^{4} arctan ( x - 2 ) + x ^{3} - 8 x ^{3} arctan ( x - 2 ) + 26 x ^{2} arctan ( x - 2 ) - 8 x ^{2} + 21 x - 40 x arctan ( x - 2 ) + 25 arctan ( x - 2 ) - 20}{10 ( x ^{2} - 4 x + 5 ) ^{2}} + 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