Find the minimum value of $ \int_0^1 (1+x^2)f^2(x)dx$ I was trying to solve a question of an entrance exam. I have taken help for a similar problem from MSE but again I am stuck in the problem. Please help me.
Let $ D= \{f \in [0,1] : f \text{continuous and} \displaystyle \int_0^1f(x)dx = 1\}$. Find the value of $$\text{min}_{f\in D} \displaystyle \int_0^1 (1+x^2)f^2(x)dx$$ 
I have tried the same eay as described in this problem but I failed to apply Euler-Lagrange equation.I can not find any other way to proceed. 
Please help me. Thnx in advance.

Here's a way:
Write $F(x) = \int_0^x f(t)dt$. 
$D$ now corresponds to those $F$ for which $F \in \mathcal C^1$, $F(0)=0$ and $F(1)=1$. And you're looking for $\min_F \int_0^1(1+x^2)(F')^2(x)dx$.
Your Lagrangian can be $\mathcal L = (1+x^2)(y')^2$. Can you take it from there? 
