1. ## Natural Logarithm Integration

Integrate (1/x+2), where x= -6, x= -3

2. Originally Posted by creatively12
Integrate (1/x+2), where x= -6, x= -3

I assume you're asking to find

$\displaystyle \int_{-6}^{-3}{\frac{1}{x + 2}\,dx}$

$\displaystyle = \left[\ln{|x + 2|}\right]_{-6}^{-3}$

$\displaystyle = \ln{|-6 + 2|} - \ln{|-3 + 2|}$

$\displaystyle = \ln{|-4|} - \ln{|-1|}$

$\displaystyle = \ln{4} - \ln{1}$

$\displaystyle = \ln{(2^2)} - 0$

$\displaystyle = 2\ln{2}$.

3. Thanks man, but the answer is -1n(4)

4. The 3 line is
$\displaystyle \ln | -3+2 | - \ln | -6 +2 |$

5. Let $\displaystyle u = x+2$, then $\displaystyle \dfrac{du}{dx} = 1$ $\displaystyle \Rightarrow {dx} = {du}$. When $\displaystyle x = -3$, $\displaystyle u = -1$; when $\displaystyle x = -6$, $\displaystyle u = -4$. So $\displaystyle \int_{-6}^{-3}\dfrac{1}{x+2}\;{dx} = \int_{-4}^{-1}\dfrac{1}{u}\;{du} = \bigg[\ln{|u|}\bigg]_{-4}^{-1} = \ln\left(1\right)-\ln\left(4\right) = \boxed{-\ln\left(4\right)}$, since $\displaystyle \ln\left(1\right) = 0$.