What is the integral of (x+3)/(x+6) ?

The integral of (x+3)/(x+6) is -3ln|x+6|+x+C

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integral of (x+3)/(x+6)

\int \frac{x + 3}{x + 6} d x

- 3 ln ∣ x + 6 ∣ + x + C

Solution steps

\int \frac{x + 3}{x + 6} d x

\frac{x + 3}{x + 6} = - \frac{3}{x + 6} + 1

= \int - \frac{3}{x + 6} + 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{3}{x + 6} d x + \int 1 d x

\int \frac{3}{x + 6} d x = 3 ln ∣ x + 6 ∣

\int 1 d x = x

= - 3 ln ∣ x + 6 ∣ + x

Add a constant to the solution

= - 3 ln ∣ x + 6 ∣ + x + C

x \to 0 + lim (\frac{2}{1 + ln ( x )})

f (x) = x, at x = 2

\frac{d y}{d t} + e^{t} y = - 3 e^{t}

sin (t) cos (t)

What is the integral of (x+3)/(x+6) ?
The integral of (x+3)/(x+6) is -3ln|x+6|+x+C