What is the integral of (x-3)/(x^2+7x+10) ?

The integral of (x-3)/(x^2+7x+10) is -5/3 ln|x+2|+8/3 ln|x+5|+C

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integral of (x-3)/(x^2+7x+10)

\int \frac{x - 3}{x ^{2} + 7 x + 10} d x

- \frac{5}{3} ln ∣ x + 2 ∣ + \frac{8}{3} ln ∣ x + 5 ∣ + C

Solution steps

\int \frac{x - 3}{x ^{2} + 7 x + 10} d x

Take the partial fraction of

\frac{x - 3}{x ^{2} + 7 x + 10} : - \frac{5}{3 ( x + 2 )} + \frac{8}{3 ( x + 5 )}

= \int - \frac{5}{3 ( x + 2 )} + \frac{8}{3 ( x + 5 )} 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{5}{3 ( x + 2 )} d x + \int \frac{8}{3 ( x + 5 )} d x

\int \frac{5}{3 ( x + 2 )} d x = \frac{5}{3} ln ∣ x + 2 ∣

\int \frac{8}{3 ( x + 5 )} d x = \frac{8}{3} ln ∣ x + 5 ∣

= - \frac{5}{3} ln ∣ x + 2 ∣ + \frac{8}{3} ln ∣ x + 5 ∣

Add a constant to the solution

= - \frac{5}{3} ln ∣ x + 2 ∣ + \frac{8}{3} ln ∣ x + 5 ∣ + C

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What is the integral of (x-3)/(x^2+7x+10) ?
The integral of (x-3)/(x^2+7x+10) is -5/3 ln|x+2|+8/3 ln|x+5|+C