Sign up to access problem solutions.

Already have an account? Log in here.

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 \(I^{-1}?\)

Sign up to access problem solutions.

Already have an account? Log in here.

Sign up to access problem solutions.

Already have an account? Log in here.

Sign up to access problem solutions.

Already have an account? Log in here.

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 \(a + b + c.\)

Sign up to access problem solutions.

Already have an account? Log in here.

Sign up to access problem solutions.

Already have an account? Log in here.

×

Problem Loading...

Note Loading...

Set Loading...