Improper integrals

In this activity we explore the improper integrals $\int_1^{\infty} \frac{1}{x} \, dx$ and $\int_1^{\infty} \frac{1}{x^{3/2}} \, dx$.

1. First we investigate $\int_1^{\infty} \frac{1}{x} \, dx$.

   1. Use the First FTC to determine the exact values of $\int_1^{10} \frac{1}{x} \, dx$, $\int_1^{1000} \frac{1}{x} \, dx$, and $\int_1^{100000} \frac{1}{x} \, dx$. Then, use your computational device to compute a decimal approximation of each result.

   2. Use the First FTC to evaluate the definite integral $\int_1^{b} \frac{1}{x} \, dx$ (which results in an expression that depends on $b$).

   3. Now, use your work from (ii.) to evaluate the limit given by .

2. N…

Determine whether each of the following improper integrals converges or diverges. For each integral that converges, find its exact value.

1. $\int_1^{\infty} \frac{1}{x^2} \, dx$

2. $\int_0^{\infty} e^{-x/4} \, dx$

3. $\int_2^{\infty} \frac{9}{(x+5)^{2/3}} \, dx$

4. $\int_4^{\infty} \frac{3}{(x+2)^{5/4}} \, dx$

5. $\int_0^{\infty} x e^{-x/4} \, dx$

6. $\int_1^{\infty} \frac{1}{x^p} \, dx$, where $p$ is a positive real number

For each of the following definite integrals, decide whether the integral is improper or not. If the integral is proper, evaluate it using the First FTC. If the integral is improper, determine whether or not the integral converges or diverges; if the integral converges, find its exact value.

1. $\int_0^1 \frac{1}{x^{1/3}} \, dx$

2. $\int_0^2 e^{-x} \, dx$

3. $\int_1^4 \frac{1}{\sqrt{4-x}} \, dx$

4. $\int_{-2}^2 \frac{1}{x^2} \, dx$

5. $\int_0^{\pi/2} \tan(x) \, dx$

6. $\int_0^1 \frac{1}{\sqrt{1-x^2}} \, dx$

Determine, with justification, whether each of the following improper integrals converges or diverges.

1. $\int_e^{\infty} \frac{\ln(x)}{x} \, dx$

2. $\int_e^{\infty} \frac{1}{x\ln(x)} \, dx$

3. $\int_e^{\infty} \frac{1}{x(\ln(x))^2} \, dx$

4. $\int_e^{\infty} \frac{1}{x(\ln(x))^p} \, dx$, where $p$ is a positive real number

5. $\int_0^1 \frac{\ln(x)}{x} \, dx$

6. $\int_0^1 \ln(x) \, dx$

Sometimes we may encounter an improper integral for which we cannot easily evaluate the limit of the corresponding proper integrals. For instance, consider $\int_1^{\infty} \frac{1}{1+x^3} \, dx$. While it is hard (or perhaps impossible) to find an antiderivative for $\frac{1}{1+x^3}$, we can still determine whether or not the improper integral converges or diverges by comparison to a simpler one. Observe that for all $x \gt 0$, $1 + x^3 \gt x^3$, and therefore .

It therefor…