What is the integral of (-x-28)/((x-2)(x^2-2x+10)) ?

The integral of (-x-28)/((x-2)(x^2-2x+10)) is -3ln|x-2|+3/2 ln|x^2-2x+10|+2/3 arctan(1/3 (x-1))+C

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integral of (-x-28)/((x-2)(x^2-2x+10))

\int \frac{- x - 28}{( x - 2 ) ( x ^{2} - 2 x + 10 )} d x

- 3 ln ∣ x - 2 ∣ + \frac{3}{2} ln x^{2} - 2 x + 10 + \frac{2}{3} arctan (\frac{1}{3} (x - 1)) + C

Solution steps

\int \frac{- x - 28}{( x - 2 ) ( x ^{2} - 2 x + 10 )} d x

Take the partial fraction of

\frac{- x - 28}{( x - 2 ) ( x ^{2} - 2 x + 10 )} : - \frac{3}{x - 2} + \frac{3 x - 1}{x ^{2} - 2 x + 10}

= \int - \frac{3}{x - 2} + \frac{3 x - 1}{x ^{2} - 2 x + 10} 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}{x - 2} d x + \int \frac{3 x - 1}{x ^{2} - 2 x + 10} d x

\int \frac{3}{x - 2} d x = 3 ln ∣ x - 2 ∣

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

= - 3 ln ∣ x - 2 ∣ + \frac{3}{2} ln x^{2} - 2 x + 10 + \frac{2}{3} arctan (\frac{1}{3} (x - 1))

Add a constant to the solution

= - 3 ln ∣ x - 2 ∣ + \frac{3}{2} ln x^{2} - 2 x + 10 + \frac{2}{3} arctan (\frac{1}{3} (x - 1)) + C

What is the integral of (-x-28)/((x-2)(x^2-2x+10)) ?
The integral of (-x-28)/((x-2)(x^2-2x+10)) is -3ln|x-2|+3/2 ln|x^2-2x+10|+2/3 arctan(1/3 (x-1))+C