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The formula for π\pi as an infinite product was derived by English mathematician John Wallis in 1655. This product, called the Wallis Product, appeared in his book Arithmetica Infinitorum.

π2=(2213)(4435)(6657)((2n)(2n)(2n1)(2n+1))\frac{\pi}{2}=\left(\frac{2 \cdot 2}{1 \cdot 3}\right)\left(\frac{4 \cdot 4}{3 \cdot 5}\right)\left(\frac{6 \cdot 6}{5 \cdot 7}\right) \cdots\left(\frac{(2 n) \cdot(2 n)}{(2 n-1) \cdot(2 n+1)}\right) \cdots

In 2015, physicists Carl Hagen and Tamar Friedmann (also a mathematician) stumbled upon a connection between quantum mechanics and the Wallis Product when they applied the variational principle to higher energy states of the hydrogen atom. This principle was previously used only on the ground energy state. The Wallis Product appeared naturally in the midst of their calculations involving gamma functions.

Quantum mechanics is the study of matter and light on the atomic and subatomic scale.

Consider Wallis's method of finding a formula for π\pi. Let

I(n)=0π/2sinnxdxI(n)=\int_0^{\pi / 2} \sin ^n x d x

From Wallis's Formulas, I(n)=(12)(34)(56)(n1n)(π2),nI(n)=\left(\frac{1}{2}\right)\left(\frac{3}{4}\right)\left(\frac{5}{6}\right) \cdots\left(\frac{n-1}{n}\right)\left(\frac{\pi}{2}\right), n is even (n2)(n \geq 2)


I(n)=(23)(45)(67)(n1n),nI(n)=\left(\frac{2}{3}\right)\left(\frac{4}{5}\right)\left(\frac{6}{7}\right) \cdots\left(\frac{n-1}{n}\right), n is odd (n3)(n \geq 3)

Find I(n)I(n) for n=2,3,4n=2,3,4, and 5 . What do you observe?


Which, if any, of the integrals listed below can be found using the 20 basic integration rules? For any that can be found, do so. For any that cannot, explain why not.

3x1x2dx\int \frac{3 x}{\sqrt{1-x^2}} d x


Answered 10 months ago
Answered 10 months ago
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We need to determine if


can be found using the 2020 basic integration rules then find it if so.

How can we determine if we can use any of the basic integration rules?

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