Let $I_1 = \int_0^1 \frac{x^2}{x^4+1}dx$ and $I_2 = \int_0^1 \frac{\log(x+\frac{1}{x})}{1+x^2}dx$, then
Step-by-Step Solution
Key Concept: Using substitution with $\cos x = t$ to transform the integral and establish relationships between $I_1$ and $I_2$
In $I_2$, let $\cos x = t$, then $(zcosax + \sin x - \sin x)dx = dt or xzcosax - dt$. Therefore $I_2 = \int_{\frac{2}{3}}^0 \frac{dt}{t} = -I_1$. Also, $I_1$ and $I_2$ are both positive as $\int_0^{1,\frac{2}{3}} > 0 \forall t \in (1, \frac{2}{3})$. Therefore the answer is $1$.
Correct Answer: 1