Calculus
# Integration Techniques

$\int_0^1 \left(1-x^2\right)^9 x^9 \, dx$

Let $I$ denote the value of the integral above. What is the sum of digits of $I^{-1}?$

Suppose a light source is positioned in $\textbf{R}^{2}$ at the origin $(0,0).$ Smooth mirrors are positioned along the lines $y = 0$ and $y = 2$ from $x = 0$ to $x = 10.$ The light source is then oriented so that the beam of light emanating from it makes an angle, chosen uniformly and at random, of between $15^{\circ}$ and $75^{\circ}$ with the positive $x$-axis. The beam is then allowed to reflect back and forth between the two mirrors until it "exits the tunnel", that is, it crosses the line $x = 10.$

If the expected distance that the beam of light travels from its source until it exits the tunnel can be expressed as $\dfrac{10}{\pi}\ln(a + b\sqrt{c}),$ where $a,b,c$ are positive integers with $c$ square-free, then find $a + b + c.$