What is the integral of (x^2)/(x^2-6x+10) ?

The integral of (x^2)/(x^2-6x+10) is 3ln|x^2-6x+10|+8arctan(x-3)+x+C

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integral of (x^2)/(x^2-6x+10)

\int \frac{x ^{2}}{x ^{2} - 6 x + 10} d x

3 ln x^{2} - 6 x + 10 + 8 arctan (x - 3) + x + C

Solution steps

\int \frac{x ^{2}}{x ^{2} - 6 x + 10} d x

\frac{x ^{2}}{x ^{2} - 6 x + 10} = \frac{6 x - 10}{x ^{2} - 6 x + 10} + 1

= \int \frac{6 x - 10}{x ^{2} - 6 x + 10} + 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{6 x - 10}{x ^{2} - 6 x + 10} d x + \int 1 d x

\int \frac{6 x - 10}{x ^{2} - 6 x + 10} d x = 3 ln x^{2} - 6 x + 10 + 8 arctan (x - 3)

\int 1 d x = x

= 3 ln x^{2} - 6 x + 10 + 8 arctan (x - 3) + x

Add a constant to the solution

= 3 ln x^{2} - 6 x + 10 + 8 arctan (x - 3) + x + C

What is the integral of (x^2)/(x^2-6x+10) ?
The integral of (x^2)/(x^2-6x+10) is 3ln|x^2-6x+10|+8arctan(x-3)+x+C