Writing an integral down is only the first step. A toolkit of techniques can help find its value, from substitutions to trigonometry to partial fractions to differentiation.

\[ \large \int_0^1 \left(1-x^2\right)^9 x^9 \, dx \]

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 \(a + b + c.\)

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