Can anyone help me in this exercise?
Be $f: \mathbb{R}^2 \to \mathbb{R}$ $$f(x, y) = \int_x^y \dfrac{t-1}{1+t^6}\ \text{d}t$$ Determine the critical point(s) of $f$ and classify it/them.
Be then $A = \{(x, y)\in\mathbb{R}^2: |x| \leq 1, |y| \leq 1\}$. Determine $f(A)$
I tried to solve the integral but there is no simple way to do it, I think. I tried to see the denominator as a sum of cubes
$$t^6+1 = (t^2)^3 + 1^3 = ((t^2+1)(t^4-t^2+1)$$ but it did not lead me to a suitable solution. Partial fractions seem no the case either. How could I do it?
Also the second request: can someone explain what does it mean $f(A)$? I really cannot understand how to proceed for this. How can I write the "function of a set"?
EDIT
My first question was really silly. Thanks to the two people who commented, I really remembered the differentiating under the integral sign theorem!
Yet I need some help for the second point...