Explain how to solve a basic equation obtained in a partial fraction decomposition that involves quadratic factors.

What does it mean for an improper integral to converge?

Describe the technique for finding $\int \sec ^{5} x \tan ^{7} x dx$. Do not integrate.

Describe how to choose u and dv when using integration by parts.

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 $(n \geq 3)$

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.

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

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.

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 $n \geq 2$.

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?

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. ∫_1^∞ [3/∛x dx]