What is the integral of (x+3)/(x^2-6x+5) ?

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

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integral of (x+3)/(x^2-6x+5)

\int \frac{x + 3}{x ^{2} - 6 x + 5} d x

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

Solution steps

\int \frac{x + 3}{x ^{2} - 6 x + 5} d x

Complete the square

x^{2} - 6 x + 5 : (x - 3)^{2} - 4

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

Apply u-substitution

= \int \frac{u + 6}{u ^{2} - 4} d u

Expand

\frac{u + 6}{u ^{2} - 4} : \frac{u}{u ^{2} - 4} + \frac{6}{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{u}{u ^{2} - 4} d u + \int \frac{6}{u ^{2} - 4} d u

\int \frac{u}{u ^{2} - 4} d u = \frac{1}{2} ln u^{2} - 4

\int \frac{6}{u ^{2} - 4} d u = - 3 (\frac{1}{2} ln \frac{u}{2} + 1 - \frac{1}{2} ln \frac{u}{2} - 1)

= \frac{1}{2} ln u^{2} - 4 - 3 (\frac{1}{2} ln \frac{u}{2} + 1 - \frac{1}{2} ln \frac{u}{2} - 1)

Substitute back

u = x - 3

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

Simplify

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

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

Add a constant to the solution

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

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