Calculus

Integration Techniques: Level 5 Challenges

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

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

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

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 $$\dfrac{\pi^a}{b}$$ for positive integers $$a,b$$, then what is the value of $$a+b$$?

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