# What is the net area between f(x) = 2/(x+3)  and the x-axis over x in [1, 2 ]?

Mar 2, 2016

$\ln \left(\frac{25}{16}\right)$

#### Explanation:

The area between the $x$ axis and our function can be given by the integral of the function $f \left(x\right)$ with the appropriate limits applied to the integral:

${\int}_{1}^{2} f \left(x\right) \mathrm{dx} = {\int}_{1}^{2} \frac{2}{x + 3} \mathrm{dx}$

We can read the integral from a table of standard integrals to obtain:

$= {\left[2 \ln | x + 3 |\right]}_{1}^{2}$

Now applying the limits:

$= \left\{2 \ln | 2 + 3 |\right\} - \left\{2 \ln | 1 + 3 |\right\}$

$= 2 \left(\ln 5 - \ln 4\right)$

Applying our rules of logarithms to do a bit of tidying up and we get:

$= \ln \left(\frac{25}{16}\right)$