Integration Techniques

Integration Techniques: Level 5 Challenges


01(1x2)9x9dx \int_0^1 \left(1-x^2\right)^9 x^9 \, dx

Let II denote the value of the integral above. What is the sum of digits of I1?I^{-1}?

0ex2dx0ex2cos2xdx= ?\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 {fn(x)}\{f_n(x)\} as follows: f1(x)=1,fn+1(x)=0etxfn(t)dt.f_1(x) = 1, \hspace{.4cm} f_{n+1}(x) = \int_0^{\infty}\frac{e^{-tx}}{f_n(t)}dt. Evaluate: f20(1)f21(1)f18(1)f19(1)\frac{f_{20}(1)f_{21}(1)}{f_{18}(1)f_{19}(1)}

Suppose a light source is positioned in R2\textbf{R}^{2} at the origin (0,0).(0,0). Smooth mirrors are positioned along the lines y=0y = 0 and y=2y = 2 from x=0x = 0 to x=10.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 1515^{\circ} and 7575^{\circ} with the positive xx-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.x = 10.

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

A function f(x)f(x) satisfies the functional equation f(tanθ)=sin22θ4f(\tan \theta)=\frac{\sin^2 2\theta}{4} for all real θ\theta. If 0f(x) dx\displaystyle \int_{0}^{\infty}f(x) \ dx is equal to πab\dfrac{\pi^a}{b} for positive integers aa and bb, then what is the value of a+ba+b?


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