What is the integral of (3x-2)/(x^2+8x+7) ?

The integral of (3x-2)/(x^2+8x+7) is 23/6 ln|x+7|-5/6 ln|x+1|+C

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integral of (3x-2)/(x^2+8x+7)

\int \frac{3 x - 2}{x ^{2} + 8 x + 7} d x

\frac{23}{6} ln ∣ x + 7 ∣ - \frac{5}{6} ln ∣ x + 1 ∣ + C

Solution steps

\int \frac{3 x - 2}{x ^{2} + 8 x + 7} d x

Expand

\frac{3 x - 2}{x ^{2} + 8 x + 7} : \frac{3 x}{x ^{2} + 8 x + 7} - \frac{2}{x ^{2} + 8 x + 7}

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{3 x}{x ^{2} + 8 x + 7} d x - \int \frac{2}{x ^{2} + 8 x + 7} d x

\int \frac{3 x}{x ^{2} + 8 x + 7} d x = 3 (\frac{7}{6} ln ∣ x + 7 ∣ - \frac{1}{6} ln ∣ x + 1 ∣)

\int \frac{2}{x ^{2} + 8 x + 7} d x = 2 (\frac{1}{6} ln ∣ x + 1 ∣ - \frac{1}{6} ln ∣ x + 7 ∣)

= 3 (\frac{7}{6} ln ∣ x + 7 ∣ - \frac{1}{6} ln ∣ x + 1 ∣) - 2 (\frac{1}{6} ln ∣ x + 1 ∣ - \frac{1}{6} ln ∣ x + 7 ∣)

Simplify

3 (\frac{7}{6} ln ∣ x + 7 ∣ - \frac{1}{6} ln ∣ x + 1 ∣) - 2 (\frac{1}{6} ln ∣ x + 1 ∣ - \frac{1}{6} ln ∣ x + 7 ∣) : \frac{23}{6} ln ∣ x + 7 ∣ - \frac{5}{6} ln ∣ x + 1 ∣

= \frac{23}{6} ln ∣ x + 7 ∣ - \frac{5}{6} ln ∣ x + 1 ∣

Add a constant to the solution

= \frac{23}{6} ln ∣ x + 7 ∣ - \frac{5}{6} ln ∣ x + 1 ∣ + C

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What is the integral of (3x-2)/(x^2+8x+7) ?
The integral of (3x-2)/(x^2+8x+7) is 23/6 ln|x+7|-5/6 ln|x+1|+C