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

Jul 31, 2017

The integral of this function cannot be evaluated with elementary functions. If you approximate it on a calculator, you'll get ${\int}_{1}^{2} f \left(x\right) \setminus \mathrm{dx} \approx - 0.94$, which will equal the "net area".

#### Explanation:

If you type "integrate (x-x^2)/ln(x^2)" into Wolfram Alpha you'll see that "nonelementary (special) functions " become involved. This not helpful for most students. Therefore, you should approximate the integral either with technology (or maybe Simpson's Rule with technology) to get ${\int}_{1}^{2} f \left(x\right) \setminus \mathrm{dx} \approx - 0.94$. Note: there is a removable singularity (hole in the graph) at $x = 1$ here, which can be "fixed" by defining $f \left(1\right) = {\lim}_{x \to 1} f \left(x\right) = - \frac{1}{2}$.

The "net (signed) area" is about $- 0.94$.

The graph can be used to confirm this visually.

graph{(x-x^2)/ln(x^2) [-10, 10, -5, 5]}