Question 40
Let $f : [\frac{1}{2}, 2] \to \mathbb{R}$ be a continuous and monotone function, for which the following equality holds:
$$
\int_{\frac{1}{2}}^{2} \frac{(1-x^2) \cdot f(x)}{1+x^2+x^4} \, dx = 0.
$$
Show that
$$
\int_{\frac{1}{2}}^{2} \frac{x f(x)}{1+x^2} \, dx = f(2) \cdot \ln 2.
$$
$$
\int_{\frac{1}{2}}^{2} \frac{(1-x^2) \cdot f(x)}{1+x^2+x^4} \, dx = 0.
$$
Show that
$$
\int_{\frac{1}{2}}^{2} \frac{x f(x)}{1+x^2} \, dx = f(2) \cdot \ln 2.
$$