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Integration Techniques

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.

Level 5

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

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

\[\large\left\lfloor \frac{\displaystyle{ \int_{0}^{\infty} e^{-x^2}\, dx}}{\displaystyle{\int_{0}^{\infty} e^{-x^2} \cos 2x \, dx}}\right\rfloor= \ ? \]

Define a sequence of functions \(\{f_n(x)\}\) as follows: \[f_1(x) = 1, \hspace{.4cm} f_{n+1}(x) = \int_0^{\infty}\frac{e^{-tx}}{f_n(t)}dt.\] Evaluate: \[\frac{f_{20}(1)f_{21}(1)}{f_{18}(1)f_{19}(1)}\]

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 \(a + b + c.\)

A function \(f(x)\) satisfies the functional equation \[f(\tan \theta)=\frac{\sin^2 2\theta}{4}\] for all real \(\theta\). If \(\displaystyle \int_{0}^{\infty}f(x) \ dx\) is equal to \(\frac{π^a}{b}\) for positive integers \(a,b\), then what is the value of \(a+b\)?

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