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

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

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

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \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

= 2 \cdot \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

= 2 (\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

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

Add a constant to the solution

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

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