What is the integral of (x-11)/(x^2+6x+25) ?

The integral of (x-11)/(x^2+6x+25) is 1/2 ln|x^2+6x+25|-7/2 arctan(1/4 (x+3))+C

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integral of (x-11)/(x^2+6x+25)

\int \frac{x - 11}{x ^{2} + 6 x + 25} d x

\frac{1}{2} ln x^{2} + 6 x + 25 - \frac{7}{2} arctan (\frac{1}{4} (x + 3)) + C

Solution steps

\int \frac{x - 11}{x ^{2} + 6 x + 25} d x

Expand

\frac{x - 11}{x ^{2} + 6 x + 25} : \frac{x}{x ^{2} + 6 x + 25} - \frac{11}{x ^{2} + 6 x + 25}

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} + 6 x + 25} d x - \int \frac{11}{x ^{2} + 6 x + 25} d x

\int \frac{x}{x ^{2} + 6 x + 25} d x = \frac{1}{2} ln x^{2} + 6 x + 25 - \frac{3}{4} arctan (\frac{1}{4} (x + 3))

\int \frac{11}{x ^{2} + 6 x + 25} d x = \frac{11}{4} arctan (\frac{1}{4} (x + 3))

= \frac{1}{2} ln x^{2} + 6 x + 25 - \frac{3}{4} arctan (\frac{1}{4} (x + 3)) - \frac{11}{4} arctan (\frac{1}{4} (x + 3))

Add similar elements:

- \frac{3}{4} arctan (\frac{1}{4} (x + 3)) - \frac{11}{4} arctan (\frac{1}{4} (x + 3)) = - \frac{7}{2} arctan (\frac{1}{4} (x + 3))

= \frac{1}{2} ln x^{2} + 6 x + 25 - \frac{7}{2} arctan (\frac{1}{4} (x + 3))

Add a constant to the solution

= \frac{1}{2} ln x^{2} + 6 x + 25 - \frac{7}{2} arctan (\frac{1}{4} (x + 3)) + C

What is the integral of (x-11)/(x^2+6x+25) ?
The integral of (x-11)/(x^2+6x+25) is 1/2 ln|x^2+6x+25|-7/2 arctan(1/4 (x+3))+C