What is the integral of (3x+7)/(x^2+5x+6) ?

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

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integral of (3x+7)/(x^2+5x+6)

\int \frac{3 x + 7}{x ^{2} + 5 x + 6} d x

ln ∣ x + 2 ∣ + 2 ln ∣ x + 3 ∣ + C

Solution steps

\int \frac{3 x + 7}{x ^{2} + 5 x + 6} d x

Take the partial fraction of

\frac{3 x + 7}{x ^{2} + 5 x + 6} : \frac{1}{x + 2} + \frac{2}{x + 3}

= \int \frac{1}{x + 2} + \frac{2}{x + 3} 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{1}{x + 2} d x + \int \frac{2}{x + 3} d x

\int \frac{1}{x + 2} d x = ln ∣ x + 2 ∣

\int \frac{2}{x + 3} d x = 2 ln ∣ x + 3 ∣

= ln ∣ x + 2 ∣ + 2 ln ∣ x + 3 ∣

Add a constant to the solution

= ln ∣ x + 2 ∣ + 2 ln ∣ x + 3 ∣ + C

\int (3 x - 8)^{21} d x

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\int 4 - t^{2} d t

\frac{d}{d y} (4 y - 1)

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