CLP-2 Integral Calculus

4 (✳)

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.)

If \(\displaystyle\int_{1}^{\infty} f(x) \,\dee{x}\) converges and \(g(x)\ge f(x)\ge 0\) for all \(x\text{,}\) then \(\displaystyle\int_{1}^{\infty} g(x) \,\dee{x}\) converges.

5

Let \(f(x) = e^{-x}\) and \(g(x)=\dfrac{1}{x+1}\text{.}\) 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.

1. \(h(x)\text{,}\) continuous and defined for all \(x \ge0\text{,}\) \(h(x) \leq f(x)\text{.}\)
2. \(h(x)\text{,}\) continuous and defined for all \(x\ge 0\text{,}\) \(f(x) \leq h(x) \leq g(x)\text{.}\)
3. \(h(x)\text{,}\) continuous and defined for all \(x\ge 0\text{,}\) \(-2f(x) \leq h(x) \leq f(x)\text{.}\)

7 (✳)

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.

8 (✳)

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

10

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

12

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

14 (✳)

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

1. \(\displaystyle\int_{-\infty}^{+\infty}\frac{x}{x^2+1}\dee{x}\) diverges
2. \(\displaystyle\int_{-\infty}^{+\infty}\frac{x}{x^2+1}\dee{x}\) converges but \(\displaystyle\int_{-\infty}^{+\infty}\left|\frac{x}{x^2+1}\right|\dee{x}\) diverges
3. \(\displaystyle\int_{-\infty}^{+\infty}\frac{x}{x^2+1}\dee{x}\) converges, as does \(\displaystyle\int_{-\infty}^{+\infty}\left|\frac{x}{x^2+1}\right|\dee{x}\)

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.

15 (✳)

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

18 (✳)

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

20

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

21

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?

23 (✳)

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

25 (✳)

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 \(10^{-4}\text{.}\) 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.

27

True or false:

There is some real number \(x\text{,}\) with \(x \geq 1\text{,}\) such that \(\displaystyle\int_0^x \frac{1}{e^t} \dee{t} = 1\text{.}\)