1.12 Improper Integrals

For which values of $b$ is the integral $\displaystyle\int_0^b \frac{1}{x^2-1}~\dee{x}$ improper?

For which values of $b$ is the integral $\displaystyle\int_0^b \frac{1}{x^2+1}~\dee{x}$ improper?

Below are the graphs $y=f(x)$ and $y=g(x)$. Suppose $\displaystyle\int_0^\infty f(x)~\dee{x}$ converges, and $\displaystyle\int_0^\infty g(x)~\dee{x}$ diverges. Assuming the graphs continue on as shown as $x \to \infty$, which graph is $f(x)$, and which is $g(x)$?

Decide whether the following statement is true or false. If false, provide a counterexample. If true, provide a brief justification. (Assume that $f(x)$ and $g(x)$ are continuous functions.)

Let $f(x) = e^{-x}$ and $g(x)=\dfrac{1}{x+1}$. Note $\int_{0}^\infty f(x)~\dee{x}$ converges while $\int_{0}^\infty g(x)~\dee{x}$ diverges.

For each of the functions $h(x)$ described below, decide whether $\int_{0\vphantom{\frac12}}^\infty h(x)~\dee{x}$ converges or diverges, or whether there isn't enough information to decide. Justify your decision.

Evaluate the integral $\displaystyle\int_0^1\frac{x^4}{x^5-1}\,\dee{x}$ or state that it diverges.

Determine whether the integral $\displaystyle\int_{-2}^2\frac{1}{(x+1)^{4/3}}\,\dee{x}$ is convergent or divergent. If it is convergent, find its value.

Does the improper integral $\displaystyle\int_1^\infty\frac{1}{\sqrt{4x^2-x}}\,\dee{x}$ converge? Justify your answer.

Does the integral $\displaystyle\int_0^\infty\frac{\dee{x}}{x^2+\sqrt{x}}$ converge or diverge? Justify your claim.

Does the integral $\displaystyle\int_{-\infty}^\infty \cos x~\dee{x}$ converge or diverge? If it converges, evaluate it.

Does the integral $\displaystyle\int_{-\infty}^\infty \sin x~\dee{x}$ converge or diverge? If it converges, evaluate it.

Evaluate $\displaystyle\int_{10}^\infty \frac{x^4-5x^3+2x-7}{x^5+3x+8} ~\dee{x}$, or state that it diverges.

Evaluate $\displaystyle\int_0^{10} \frac{x-1}{x^2-11x+10} ~\dee{x}$, or state that it diverges.

Determine (with justification!) which of the following applies to the integral $\displaystyle\int_{-\infty}^{+\infty}\frac{x}{x^2+1}\dee{x}$:

Remark: these options, respectively, are that the integral diverges, converges conditionally, and converges absolutely. You'll see this terminology used for series in Section 3.4.1 of the CLP-2 text.

Decide whether $I=\displaystyle\int_0^\infty\frac{|\sin x|}{x^{3/2}+x^{1/2}}\ \dee{x}$ converges or diverges. Justify.

Does the integral $\displaystyle\int_0^\infty\frac{x+1}{x^{1/3}(x^2+x+1)}\,\dee{x}$ converge or diverge?

We craft a tall, vuvuzela-shaped solid by rotating the line $y = \dfrac{1}{x\vphantom{\frac{1}{2}}}$ from $x=a$ to $x=1$ about the $y$-axis, where $a$ is some constant between 0 and 1.

True or false: No matter how large a constant $M$ is, there is some value of $a$ that makes a solid with volume larger than $M$.

What is the largest value of $q$ for which the integral $\displaystyle \int_1^\infty \frac1{x^{5q}}\,\dee{x}$ diverges?

For which values of $p$ does the integral $\displaystyle\int_0^\infty \dfrac{x}{(x^2+1)^p}~\dee{x}$ converge?

Evaluate $\displaystyle\int_2^\infty \frac{1}{t^4-1}\dee{t}$, or state that it diverges.

Does the integral $\displaystyle\int_{-5}^5 \left(\frac{1}{\sqrt{|x|}} + \frac{1}{\sqrt{|x-1|}}+\frac{1}{\sqrt{|x-2|}}\right)\dee{x}$ converge or diverge?

Evaluate $\displaystyle\int_0^\infty e^{-x}\sin x~\dee{x}$, or state that it diverges.

Is the integral $\displaystyle\int_0^\infty\frac{\sin^4 x}{x^2}\, \dee{x}$ convergent or divergent? Explain why.

Does the integral $\displaystyle\int_0^\infty \frac{x}{e^x+\sqrt{x}} ~\dee{x}$ converge or diverge?

Let $M_{n,t}$ be the Midpoint Rule approximation for $\displaystyle\int_0^t \frac{e^{-x}}{1+x}\ \dee{x}$ with $n$ equal subintervals. Find a value of $t$ and a value of $n$ such that $M_{n,t}$ differs from $\int_0^\infty \frac{e^{-x}}{1+x}\ \dee{x}$ by at most $10^{-4}$. Recall that the error $E_n$ introduced when the Midpoint Rule is used with $n$ subintervals obeys

where $M$ is the maximum absolute value of the second derivative of the integrand and $a$ and $b$ are the end points of the interval of integration.

Suppose $f(x)$ is continuous for all real numbers, and $\displaystyle\int_1^\infty f(x)~\dee{x}$ converges.

True or false: There is some real number $x$, with $x \geq 1$, such that $\displaystyle\int_0^x \frac{1}{e^t}~\dee{t} = 1$.