Question

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

Answer 1

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(x)\ 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(x)\ 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(1)=lim_{x->1}f(x)=-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]}