What is the integral of (2x-1)/(x^2-6x+13) ?

The integral of (2x-1)/(x^2-6x+13) is ln|x^2-6x+13|+5/2 arctan(1/2 (x-3))+C

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integral of (2x-1)/(x^2-6x+13)

\int \frac{2 x - 1}{x ^{2} - 6 x + 13} d x

ln x^{2} - 6 x + 13 + \frac{5}{2} arctan (\frac{1}{2} (x - 3)) + C

Solution steps

\int \frac{2 x - 1}{x ^{2} - 6 x + 13} d x

Complete the square

x^{2} - 6 x + 13 : (x - 3)^{2} + 4

= \int \frac{2 x - 1}{( x - 3 ) ^{2} + 4} d x

Apply u-substitution

= \int \frac{2 u + 5}{u ^{2} + 4} d u

Expand

\frac{2 u + 5}{u ^{2} + 4} : \frac{2 u}{u ^{2} + 4} + \frac{5}{u ^{2} + 4}

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{2 u}{u ^{2} + 4} d u + \int \frac{5}{u ^{2} + 4} d u

\int \frac{2 u}{u ^{2} + 4} d u = ln u^{2} + 4

\int \frac{5}{u ^{2} + 4} d u = \frac{5}{2} arctan (\frac{u}{2})

= ln u^{2} + 4 + \frac{5}{2} arctan (\frac{u}{2})

Substitute back

u = x - 3

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

Simplify

ln (x - 3)^{2} + 4 + \frac{5}{2} arctan (\frac{x - 3}{2}) : ln x^{2} - 6 x + 13 + \frac{5}{2} arctan (\frac{1}{2} (x - 3))

= ln x^{2} - 6 x + 13 + \frac{5}{2} arctan (\frac{1}{2} (x - 3))

Add a constant to the solution

= ln x^{2} - 6 x + 13 + \frac{5}{2} arctan (\frac{1}{2} (x - 3)) + C

What is the integral of (2x-1)/(x^2-6x+13) ?
The integral of (2x-1)/(x^2-6x+13) is ln|x^2-6x+13|+5/2 arctan(1/2 (x-3))+C