Question #41801

Let $X$ be a continuous random variable taking values in the interval $[-1, 1]$. Its probability density function $f(x)$ satisfies the integral equation: $\int_{0}^{x} f(t) \sqrt{1+t^2} \, dt = \frac{1}{4} \left( x \sqrt{1+x^2} + \ln(x + \sqrt{1+x^2}) \right) + \frac{1}{6} \left( (1+x^2)^{3/2} - 1 \right)$ for all $x \in [-1, 1]$. Let $E_1$ be the event that the roots of the quadratic equation $y^2 + 2Xy + X = 0$ (in variable $y$) are real. Let $E_2$ be the event that $X$ satisfies the inequality: $\int_{-X}^{X} \frac{t^3 \cos^2 t}{1+t^2} \, dt + X^2 \le \frac{1}{4}$ Find the conditional probability $P(E_1 \mid E_2)$.