What is the integral of (x+66)/(x^2-36) ?

The integral of (x+66)/(x^2-36) is 1/2 ln|x^2-36|-11(1/2 ln| x/6+1|-1/2 ln| x/6-1|)+C

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integral of (x+66)/(x^2-36)

\int \frac{x + 66}{x ^{2} - 36} d x

\frac{1}{2} ln x^{2} - 36 - 11 (\frac{1}{2} ln \frac{x}{6} + 1 - \frac{1}{2} ln \frac{x}{6} - 1) + C

Solution steps

\int \frac{x + 66}{x ^{2} - 36} d x

Expand

\frac{x + 66}{x ^{2} - 36} : \frac{x}{x ^{2} - 36} + \frac{66}{x ^{2} - 36}

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{x}{x ^{2} - 36} d x + \int \frac{66}{x ^{2} - 36} d x

\int \frac{x}{x ^{2} - 36} d x = \frac{1}{2} ln x^{2} - 36

\int \frac{66}{x ^{2} - 36} d x = - 11 (\frac{1}{2} ln \frac{x}{6} + 1 - \frac{1}{2} ln \frac{x}{6} - 1)

= \frac{1}{2} ln x^{2} - 36 - 11 (\frac{1}{2} ln \frac{x}{6} + 1 - \frac{1}{2} ln \frac{x}{6} - 1)

Add a constant to the solution

= \frac{1}{2} ln x^{2} - 36 - 11 (\frac{1}{2} ln \frac{x}{6} + 1 - \frac{1}{2} ln \frac{x}{6} - 1) + C

What is the integral of (x+66)/(x^2-36) ?
The integral of (x+66)/(x^2-36) is 1/2 ln|x^2-36|-11(1/2 ln| x/6+1|-1/2 ln| x/6-1|)+C