What is the integral of (x^2+2x+7)/(x^2+6x+5) ?

The integral of (x^2+2x+7)/(x^2+6x+5) is 3/2 ln|x+1|-11/2 ln|x+5|+x+C

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integral of (x^2+2x+7)/(x^2+6x+5)

\int \frac{x ^{2} + 2 x + 7}{x ^{2} + 6 x + 5} d x

\frac{3}{2} ln ∣ x + 1 ∣ - \frac{11}{2} ln ∣ x + 5 ∣ + x + C

Solution steps

\int \frac{x ^{2} + 2 x + 7}{x ^{2} + 6 x + 5} d x

\frac{x ^{2} + 2 x + 7}{x ^{2} + 6 x + 5} = \frac{- 4 x + 2}{x ^{2} + 6 x + 5} + 1

= \int \frac{- 4 x + 2}{x ^{2} + 6 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 + 2}{x ^{2} + 6 x + 5} d x + \int 1 d x

\int \frac{- 4 x + 2}{x ^{2} + 6 x + 5} d x = \frac{3}{2} ln ∣ x + 1 ∣ - \frac{11}{2} ln ∣ x + 5 ∣

\int 1 d x = x

= \frac{3}{2} ln ∣ x + 1 ∣ - \frac{11}{2} ln ∣ x + 5 ∣ + x

Add a constant to the solution

= \frac{3}{2} ln ∣ x + 1 ∣ - \frac{11}{2} ln ∣ x + 5 ∣ + x + C

What is the integral of (x^2+2x+7)/(x^2+6x+5) ?
The integral of (x^2+2x+7)/(x^2+6x+5) is 3/2 ln|x+1|-11/2 ln|x+5|+x+C