Why is completing the square useful when you are considering integration by trigonometric substitution?

Describe what is meant by a reduction formula. Give an example.

The formula for $\pi$ 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.

$$\frac{\pi}{2}=\left(\frac{2 \cdot 2}{1 \cdot 3}\right)\left(\frac{4 \cdot 4}{3 \cdot 5}\right)\left(\frac{6 \cdot 6}{5 \cdot 7}\right) \cdots\left(\frac{(2 n) \cdot(2 n)}{(2 n-1) \cdot(2 n+1)}\right) \cdots$$

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 $\pi$. Let

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

From Wallis's Formulas, $I(n)=\left(\frac{1}{2}\right)\left(\frac{3}{4}\right)\left(\frac{5}{6}\right) \cdots\left(\frac{n-1}{n}\right)\left(\frac{\pi}{2}\right), n$ is even $(n \geq 2)$

or

$I(n)=\left(\frac{2}{3}\right)\left(\frac{4}{5}\right)\left(\frac{6}{7}\right) \cdots\left(\frac{n-1}{n}\right), n$ is odd $(n \geq 3)$

Show that

$$\lim _{n \rightarrow \infty} \frac{I(2 n+1)}{I(2 n)}=1$$

(Hint: Use the Squeeze Theorem.)

The formula for $\pi$ 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.

$$\frac{\pi}{2}=\left(\frac{2 \cdot 2}{1 \cdot 3}\right)\left(\frac{4 \cdot 4}{3 \cdot 5}\right)\left(\frac{6 \cdot 6}{5 \cdot 7}\right) \cdots\left(\frac{(2 n) \cdot(2 n)}{(2 n-1) \cdot(2 n+1)}\right) \cdots$$

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 $\pi$. Let

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

From Wallis's Formulas, $I(n)=\left(\frac{1}{2}\right)\left(\frac{3}{4}\right)\left(\frac{5}{6}\right) \cdots\left(\frac{n-1}{n}\right)\left(\frac{\pi}{2}\right), n$ is even $(n \geq 2)$

or

$I(n)=\left(\frac{2}{3}\right)\left(\frac{4}{5}\right)\left(\frac{6}{7}\right) \cdots\left(\frac{n-1}{n}\right), n$ is odd $(n \geq 3)$

A tank contains 1000 L of pure water. Brine that contains 0.05 kg of salt per liter of water enters the tank at a rate of 5 L/min. Brine that contains 0.04 kg of salt per liter of water enters the tank at a rate of 10 L/min. The solution is kept thoroughly mixed and drains from the tank at a rate of 15 L/min. How much salt is in the tank after t minutes

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.

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

The formula for $\pi$ 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.

$$\frac{\pi}{2}=\left(\frac{2 \cdot 2}{1 \cdot 3}\right)\left(\frac{4 \cdot 4}{3 \cdot 5}\right)\left(\frac{6 \cdot 6}{5 \cdot 7}\right) \cdots\left(\frac{(2 n) \cdot(2 n)}{(2 n-1) \cdot(2 n+1)}\right) \cdots$$

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 $\pi$. Let

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

From Wallis's Formulas, $I(n)=\left(\frac{1}{2}\right)\left(\frac{3}{4}\right)\left(\frac{5}{6}\right) \cdots\left(\frac{n-1}{n}\right)\left(\frac{\pi}{2}\right), n$ is even $(n \geq 2)$

or

$I(n)=\left(\frac{2}{3}\right)\left(\frac{4}{5}\right)\left(\frac{6}{7}\right) \cdots\left(\frac{n-1}{n}\right), n$ is odd $(n \geq 3)$

Show that $I(n+1) \leq I(n)$ for $n \geq 2$.

Use Wallis’s Formulas to prove that ∫_-1^1 (1 - x²)^n dx = 2^2n+1(n!)² / (2n + 1)! for all positive integers n.

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.

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

Evaluate the integrals ∫_-1^1 (1-x²)dx and ∫_-1^1 (1-x²)² dx.

The formula for $\pi$ 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.

$$\frac{\pi}{2}=\left(\frac{2 \cdot 2}{1 \cdot 3}\right)\left(\frac{4 \cdot 4}{3 \cdot 5}\right)\left(\frac{6 \cdot 6}{5 \cdot 7}\right) \cdots\left(\frac{(2 n) \cdot(2 n)}{(2 n-1) \cdot(2 n+1)}\right) \cdots$$

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 $\pi$. Let

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

From Wallis's Formulas, $I(n)=\left(\frac{1}{2}\right)\left(\frac{3}{4}\right)\left(\frac{5}{6}\right) \cdots\left(\frac{n-1}{n}\right)\left(\frac{\pi}{2}\right), n$ is even $(n \geq 2)$

or

$I(n)=\left(\frac{2}{3}\right)\left(\frac{4}{5}\right)\left(\frac{6}{7}\right) \cdots\left(\frac{n-1}{n}\right), n$ is odd $(n \geq 3)$

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

Which of the following changes will most likely result in an increase in the demand for hamburgers in your hometown? A. The price of hotdog s decreases. B. The price of drinks sold at hamburger restaurants increases. C. Income in your town decreases and hamburgers are a normal good. D. The local newspaper publishes a story on health problems caused by red meat. E. The number of vegetarians in your town decreases and the population size remains the same.

Find the trigonometric integral. ∫ cos³(πx - 1) dx

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.

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