What is the integral of ((7x+29))/(x^2+8x+15) ?

The integral of ((7x+29))/(x^2+8x+15) is 4ln|x+3|+3ln|x+5|+C

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integral of ((7x+29))/(x^2+8x+15)

\int \frac{( 7 x + 29 )}{x ^{2} + 8 x + 15} d x

4 ln ∣ x + 3 ∣ + 3 ln ∣ x + 5 ∣ + C

Solution steps

\int \frac{7 x + 29}{x ^{2} + 8 x + 15} d x

Take the partial fraction of

\frac{7 x + 29}{x ^{2} + 8 x + 15} : \frac{4}{x + 3} + \frac{3}{x + 5}

= \int \frac{4}{x + 3} + \frac{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{4}{x + 3} d x + \int \frac{3}{x + 5} d x

\int \frac{4}{x + 3} d x = 4 ln ∣ x + 3 ∣

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

= 4 ln ∣ x + 3 ∣ + 3 ln ∣ x + 5 ∣

Add a constant to the solution

= 4 ln ∣ x + 3 ∣ + 3 ln ∣ x + 5 ∣ + C

What is the integral of ((7x+29))/(x^2+8x+15) ?
The integral of ((7x+29))/(x^2+8x+15) is 4ln|x+3|+3ln|x+5|+C