What is the integral of ((x-3))/((x-2)(x^2+1)) ?

The integral of ((x-3))/((x-2)(x^2+1)) is -1/5 ln|x-2|+1/5 (1/2 ln|x^2+1|+7arctan(x))+C

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integral of ((x-3))/((x-2)(x^2+1))

\int \frac{( x - 3 )}{( x - 2 ) ( x ^{2} + 1 )} d x

- \frac{1}{5} ln ∣ x - 2 ∣ + \frac{1}{5} (\frac{1}{2} ln x^{2} + 1 + 7 arctan (x)) + C

Solution steps

\int \frac{x - 3}{( x - 2 ) ( x ^{2} + 1 )} d x

Take the partial fraction of

\frac{x - 3}{( x - 2 ) ( x ^{2} + 1 )} : - \frac{1}{5 ( x - 2 )} + \frac{x + 7}{5 ( x ^{2} + 1 )}

= \int - \frac{1}{5 ( x - 2 )} + \frac{x + 7}{5 ( x ^{2} + 1 )} 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{1}{5 ( x - 2 )} d x + \int \frac{x + 7}{5 ( x ^{2} + 1 )} d x

\int \frac{1}{5 ( x - 2 )} d x = \frac{1}{5} ln ∣ x - 2 ∣

\int \frac{x + 7}{5 ( x ^{2} + 1 )} d x = \frac{1}{5} (\frac{1}{2} ln x^{2} + 1 + 7 arctan (x))

= - \frac{1}{5} ln ∣ x - 2 ∣ + \frac{1}{5} (\frac{1}{2} ln x^{2} + 1 + 7 arctan (x))

Add a constant to the solution

= - \frac{1}{5} ln ∣ x - 2 ∣ + \frac{1}{5} (\frac{1}{2} ln x^{2} + 1 + 7 arctan (x)) + C

What is the integral of ((x-3))/((x-2)(x^2+1)) ?
The integral of ((x-3))/((x-2)(x^2+1)) is -1/5 ln|x-2|+1/5 (1/2 ln|x^2+1|+7arctan(x))+C