What is the integral of (x^2)/(x^2-4x+5) ?

The integral of (x^2)/(x^2-4x+5) is 2ln|x^2-4x+5|+3arctan(x-2)+x+C

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integral of (x^2)/(x^2-4x+5)

\int \frac{x ^{2}}{x ^{2} - 4 x + 5} d x

2 ln x^{2} - 4 x + 5 + 3 arctan (x - 2) + x + C

Solution steps

\int \frac{x ^{2}}{x ^{2} - 4 x + 5} d x

\frac{x ^{2}}{x ^{2} - 4 x + 5} = \frac{4 x - 5}{x ^{2} - 4 x + 5} + 1

= \int \frac{4 x - 5}{x ^{2} - 4 x + 5} + 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{4 x - 5}{x ^{2} - 4 x + 5} d x + \int 1 d x

\int \frac{4 x - 5}{x ^{2} - 4 x + 5} d x = 2 ln x^{2} - 4 x + 5 + 3 arctan (x - 2)

\int 1 d x = x

= 2 ln x^{2} - 4 x + 5 + 3 arctan (x - 2) + x

Add a constant to the solution

= 2 ln x^{2} - 4 x + 5 + 3 arctan (x - 2) + x + C

What is the integral of (x^2)/(x^2-4x+5) ?
The integral of (x^2)/(x^2-4x+5) is 2ln|x^2-4x+5|+3arctan(x-2)+x+C