Date modified: Tue 2023-07-11 . 04:28 AM
Date created: Sat 2023-10-07 . 09:02 PM
# Week 4 Tuesday Problems ## Reading. Please read sections 7.7 and 7.8. Do try the exercises for additional practice. You got this!! ## Problems. 1. Here are some basic examples of improper integrals that are **very important** for intuition. Determine whether the following converge or diverges. If converge, give its value. 1. $\displaystyle\int_{1}^{\infty} \frac{1}{x}dx$ 2. $\displaystyle\int_{0}^{1} \frac{1}{x}dx$ 3. $\displaystyle\int_{1}^{\infty} \frac{1}{x^{2}}dx$ 4. $\displaystyle\int_{0}^{1} \frac{1}{x^{2}}dx$ 5. $\displaystyle\int_{1}^{\infty} \frac{1}{\sqrt{x}}dx$ 6. $\displaystyle\int_{0}^{1} \frac{1}{\sqrt{x}}dx$ 2. The following concerns the classical definition of improper integrals. 1. Does the improper integral $\displaystyle\int_{0}^{\infty}xdx$ converge or diverges? Show using its limit definition. 2. Does the improper integral $\displaystyle\int_{-\infty}^{\infty}xdx$ converge or diverge? Show using its definition. 3. Compute the definite integral $\displaystyle\int_{-t}^{t}xdx$. This should give you an expression in $t$. 4. What is the limit $\displaystyle\lim_{t\to\infty}\int_{-t}^{t}xdx$? Does this mean anything to the convergence of the improper integral $\int_{-\infty}^{\infty}xdx$? **Caution. This limit is not how we classically define improper integrals of this type!** 3. Determine if the following improper integrals converge or diverge. If converges, compute its value as well. You will need the limit definition. 1. $\displaystyle\int_{-2}^{\infty} \frac{1}{x+4}dx$ 2. $\displaystyle\int_{1}^{\infty} \frac{1}{x^{2}+4}dx$ 3. $\displaystyle\int_{0}^{\infty} e^{-2x}dx$ 4. $\displaystyle\int_{-\infty}^{-1} \frac{1}{\sqrt[3]{x}}dx$ 5. $\displaystyle\int_{-\infty}^{\infty}\cos(2x)dx$ 6. $\displaystyle\int_{1}^{\infty} \frac{e^{-1/x}}{x^{2}}dx$ 7. $\displaystyle\int_{e}^{\infty} \frac{1}{x(\ln(x))^{2}}dx$ 8. $\displaystyle\int_{1}^{\infty} \frac{1}{\sqrt{x}+x\sqrt{x}}dx$. 4. Determine if the following improper integrals converges or diverges. No need to determine the values if they converge. 1. $\displaystyle\int_{2}^{\infty} \frac{1}{x-\ln(x)}dx$ 2. $\displaystyle\int_{0}^{\infty} \frac{\arctan(x)}{2+e^{x}}dx$ 3. $\displaystyle\int_{2}^{\infty} \frac{x+1}{\sqrt{x^{4}-x}}dx$ 4. $\displaystyle\int_{1}^{2} \frac{x+1}{\sqrt{x^{4}-x}}dx$ 5. $\displaystyle\int_{0}^{1} \frac{\sec^{2}(x)}{x\sqrt{x}}dx$ 6. $\displaystyle\int_{0}^{\pi} \frac{\sin^{2}(x)}{\sqrt{x}}dx$