Questions involving improper integrals, defined as the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞, or as both endpoints approach limits.

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6
votes
0answers
57 views

Questions regarding $\int_0^\infty\frac{\sin^2x}{x^2}dx$ [duplicate]

question: $\int_0^\infty\frac{\sin^2x}{x^2}dx$ is equal to (A)$\int_0^\infty\frac{\sin x}{x}dx$ (B)$\int_0^\infty\frac{\cos x}{x}dx$ (C)$\int_0^\infty\frac{\cos^2x}{x^2}dx$ ...
0
votes
2answers
62 views

Is this integral less than infinity?

Assume the following integral: $$ \int\limits_{-\infty}^{\infty}\frac{f\left(x\right)} {BB\left(\lceil abs\left(x\right)\rceil\right)}\mathrm{d}x $$ Where $f\left(x\right)$ is any computable ...
5
votes
2answers
80 views

Improper integral of $\log x \operatorname{sech} x$

How to prove the following? $$ \int_0^\infty \log x \operatorname{sech}x\,dx = \frac{\pi}{2} \log\left( \frac{4\pi^3}{\Gamma(1/4)^4} \right) $$ I obtained the right side with CAS. It seems like this ...
0
votes
0answers
49 views

Improper integral of $\frac{\sin (\pi a t) \sin (\pi b t)}{t^2}$ depends only on the smaller of $a,b$ [duplicate]

$$\int_{-\infty }^{\infty } \frac{\sin (\pi a t) \sin (\pi b t)}{t^2} \, dt = \pi ^2 b$$ if $a$ and $b$ positive and $a>b$ (from Mathematica). How is this result only based on the smaller of ...
2
votes
1answer
24 views

Proof for the behavior of both types of improper integrals for different powers of x

I was trying to prove for what values of p eq.1 converges or diverges, they didn't give the proof for eq.1 but for eq.2 a proof was given and when I was done with the proof for eq.1 I noticed that for ...
0
votes
3answers
104 views

Properties of improper integral (showing that: $\int \limits_{0}^{\infty}f(x)dx=\int \limits_{0}^{1}f(x)dx+\int \limits_{1}^{\infty}f(x)dx.$)

Let $f(x)$ is integrable on every segment $[r,\infty)$ where $r>0$. Let $\int \limits_{0}^{1}f(x)dx$ and $\int \limits_{1}^{\infty}f(x)dx$ converges. Why in this case we can conclude that $$\int ...
0
votes
0answers
57 views

Integrate $\int^1_{0} \frac{\ln (x+1)}{x^2+1}dx$ [duplicate]

$$\int^1_{0}\frac{\ln (x+1)}{x^2+1}dx$$ I'm having trouble solving this one. I tried trigonometric subst. but that doesn't get me far:$$\int^1_{0}\frac{\ ln (tan\theta+1)}{\sec^2\theta}\sec^2\theta ...
6
votes
1answer
82 views

Is $\int_{M_{n}(\mathbb{R})} e^{-A^{2}}d\mu$ a convergent integral?(2)

We identify $M_{n}(\mathbb{R})$ with $\mathbb{R}^{n^{2}}$ We put $\int_{M_{n}(\mathbb{R})} e^{-A^{2}}d\mu=\lim_{r\to \infty} \int_{D_{r}} e^{-A^{2}}$ where the later is counted as a Riemann ...
2
votes
3answers
54 views

Continuity of improper integral with a continuous integrand.

I am a newbie in analysis and am trying to wrap my head around some continuity/compactness/finiteness concepts. Let $f(x,y):\mathbb{R}^2\mapsto\mathbb{R}$ be a continuous function in both $x$ and $y$ ...
3
votes
4answers
87 views

Can you compute $\int_0^1\frac{\log(x)\log(1-x)}{x}dx$ more precisely than $1.20206$ and do a comparision with $\zeta(3)$?

I know from Wolfram Alpha that $$\int_0^1\frac{\log(x)\log(1-x)}{x}dx=1.20206$$ and in the other hand, too from this online tool that ...
0
votes
2answers
62 views

Show $\int^1_0 \frac{\sin x -x}{x^3}$ converges.

I'm trying to work through a couple improper integral problems to study for a final, but am somewhat stuck on showing that $\displaystyle \int^1_0 \frac{\sin x -x}{x^3}$ converges. Edit: Can you use ...
10
votes
1answer
151 views

Is $\int_{M_{n}(\mathbb{R})} e^{-A^{2}}d\mu$ a convergent integral?

Is the following integral a convergent integral? Can we compute it, precisely? $$\int_{M_{n}(\mathbb{R})} e^{-A^{2}}d\mu $$ Here $\mu$ is the usual measure of $M_{n}(\mathbb{R})\simeq ...
1
vote
1answer
36 views

Integrating an absolute value on exponential

This might be a bit rusty but hopefully it can be brushed up. I need to integrate $$\int_{-\infty}^{\infty}xe^{-2\lambda \left | x \right |}dx$$ Recall: $$\left | x \right |=\left\{\begin{matrix} ...
0
votes
0answers
38 views

Integral of exponential

I have the expected value of the square of the position operator given as $$\langle x^{2} \rangle= \int_{-\infty}^{\infty} x^{2}e^{\frac{amx^{2}}{h}}dx$$ I understand that the integrand can only be ...
1
vote
1answer
39 views

Prove that the function $f(x)=\frac{\cos^2x}{\sqrt{x^4+1}}$ is improperly integrable on $(0,\infty)$.

My attempt: We know $f(x)=\frac{\cos^2x}{\sqrt{x^4+1}}$ (or rather, I am just assuming) is locally integrable and positive, and $$0\leq f(x)\leq\frac{1}{\sqrt{x^4+1}}\leq\frac{1}{x^2},$$ and we know ...
0
votes
0answers
28 views

Prove $L=0$ when improper integral of $f(x)$ converges and $\lim f(x) =L$ as $x \rightarrow \infty$

Here is the question. If $\int ^\infty _1 f(x)dx$ converges and $\lim_{x\to \infty} f(x)=L$, prove that $L=0$. Any help would be appreciated.
2
votes
0answers
37 views

Test integral convergence

I'm given the integral $$\int \limits_0 ^{+ \infty} \frac{e ^ {-\cos t} \cdot \sin (t ^ \beta)}{t^\alpha} dt \qquad a,b \in \mathbb{R}$$ and I need to test the absolute convergence. I split it in two ...
0
votes
0answers
42 views

Values for which Improper Integrals Converge

I want to determine the values of $a$ and $b$ for which the following integrals converge: (i) $$\int \limits _0 ^{\infty} \frac{x^{a-1}}{1+xb}dx$$ (ii) $$\int \limits _0^1 x^a(1-x^2)^b dx$$ My ...
2
votes
4answers
69 views

Convergence of an integral $\int_1^{+\infty} \frac{1}{x\sqrt[3]{x^2+1}}\mathrm dx$

Convergence of an integral $\int_1^{+\infty} \frac{1}{x\sqrt[3]{x^2+1}}\mathrm dx$ $\int_1^{+\infty} \frac{1}{x\sqrt[3]{x^2+1}}\mathrm dx=\lim\limits_{t\to\infty}\int_1^{t} ...
9
votes
7answers
965 views

What is the value of this improper integral? $\lim_{x\rightarrow 0 } \dfrac{1}{x} \int_{x}^{2x} e^{-t^2}\,\mathrm dt$

$$\lim_{x\rightarrow 0 } \dfrac{1}{x} \int_{x}^{2x} e^{-t^2}\,\mathrm dt$$ I don't have any idea to solve this integral.
0
votes
1answer
35 views

For which values of $\alpha$ and $\beta$ does the integral $\int\limits_2^{\infty}\frac{dx}{x^{\alpha}ln^{\beta}x}$ converge?

I'm trying to find out for which values of $\alpha$ and $\beta$ the integral $\int\limits_2^{\infty}\frac{dx}{x^{\alpha}ln^{\beta}x}$ does converge. I know that when $\alpha=1$ then $\beta$ must be ...
3
votes
2answers
47 views

Transforming the integral $\int_0^\infty e^{-x^2}\sin(x) dx$ into a specific sum

Using the series representation of $\sin x$, I want to prove that: $$\int_0^\infty e^{-x^2} \sin(x) dx = \frac{1}{2} \sum_{k=0}^\infty (-1)^k \frac{k!}{(2k+1)!}$$ My attempt: I've started by ...
0
votes
1answer
28 views

Is it possible for an improper integral to converge and its series to diverge? [closed]

Can an improper Riemann integral converge while its infinite series diverges?
1
vote
0answers
28 views

Compare $\int_{-\infty}^{+\infty} f(x) dx \quad\text{ and }\quad \lim_{t\rightarrow +\infty} \int_{-t}^{t} f(x) dx$

I would like to compare these two integrals $$\int_{-\infty}^{+\infty} f(x) dx \quad\text{ and }\quad \lim_{t\rightarrow +\infty} \int_{-t}^{t} f(x) dx$$ My Thoughts: Let $\displaystyle ...
1
vote
2answers
43 views

solution of an improper integral.

I was solving following improper integral: $$ \int\limits_0^\frac{\pi}{2}\frac{log~x}{x^a}dx $$ where $a<1$. My attempt: $0$ is the only point of discontinuity. So, $\frac{log~x}{x^a}\leq ...
0
votes
2answers
50 views

If $f$ is positive, decreasing, and continuous on $[0,+\infty)$ show that $\int_0^\infty f(t)dt$ converges iff $\sum_{n=0}^\infty f(n)$ converges

My Work: Integrals take infinitely small steps to get from one term to the next, whereas in a series, the distance between each term must be some tangible value ($|x_n-x_{n-1}|\ge \epsilon$). Thus, ...
1
vote
1answer
44 views

Is $f(x)=\frac{\sqrt{1-x}}{\ln{x}}$ on $[0,1]$ a Lebesgue-integrable function?

I have to prove that $\displaystyle x\mapsto\frac{\sqrt{1-x}}{\ln{x}}$ is Lebesgue-integrable on $[0,1]$. So I try to bound $\displaystyle\left|\frac{\sqrt{1-x}}{\ln{x}}\right|$ with a ...
0
votes
0answers
23 views

Pdf, cdf and Fundamental Theorem of Calculus

I have some doubts related to the definition/properties of the probability distribution function and the probability density function of a continuous random variable $X$. The cdf of $X$ is ...
0
votes
1answer
11 views

Splitting a complex controur integration in two. Figuring out the orientation.

Say I have an integration $$\int_{L_1} f(z)dz $$ that I want to write as a sum of $$\int_{L_2} f(z)dz \quad and \quad \int_{L_3} f(z)dz $$ $L_1,L_2$ are positively oriented. Suppose $L_3$ be ...
2
votes
3answers
111 views

Integral from $\infty$ to $\infty$

I know there is the property that $$\int_a^a f(x) \mathbb{d} x = 0$$ But what if $a=\infty$? Is the integral still necessarily zero even though infinity isn't well defined?
1
vote
1answer
39 views

Diverging integrals less than |sin(x)/x| (A Dirichlet Integral)

In my probability class we were asked to show that $\int_0^\infty \left\vert\frac{\sin(x)}{x}\right\vert = \infty$. The hint states to find a function below that is easier to integrate and show it ...
3
votes
2answers
92 views

Closed-form solution of the integral $\int_{0}^{\infty}\frac{\ln(1+x)}{(1+ax)^{(1+m)}}\,dx$

I want to know whether there is a closed form solution for the integral $$\int_{0}^{\infty}\frac{\ln(1+x)}{(1+ax)^{(1+m)}}\,dx$$ where $a$ and $m$ are positive (not necessarily integers). A ...
1
vote
2answers
38 views

Help with improper integrals containing partial fractions

Specifically, $$\int_1^\infty \frac {24}{8x(x+1)^2} dx$$ and $$\int_3^\infty \frac {1}{t^2 - 2t} dt$$ Both of these problems are supposed to converge, but I keep getting infinity in my answer. For ...
2
votes
2answers
40 views

Finding a dominating function to evaluate arctan integral

I want to find a dominating function to evaluate the limit of the integral $$ \int_{-\infty}^{\infty} \arctan\left(\frac{1}{n^2}(b - x)\right) - \arctan\left(\frac{1}{n^2}(a - x)\right) \; dx $$ as ...
0
votes
1answer
21 views

Does $\int^{\infty}_2 {e^{x/4} \over x^3({\ln x})^5} $ converge?

Does $$\int^{\infty}_2 {e^{x/4} \over x^3({\ln x})^5}dx $$ converge? I don't know how to start - What can I compare it to? It seems too complicated. How do you approach to this kind of problems? ...
1
vote
3answers
93 views

Analysis: Prove that this improper integral diverges but the limit as $t \rightarrow \infty=\pi$?

Question: Show that the improper integral $$\int_{-\infty}^{\infty}\frac{1+x}{1+x^2}dx$$ diverges but that $$\lim_{t\rightarrow\infty}\int_{-t}^{t}\frac{1+x}{1+x^2}dx=\pi.$$
1
vote
3answers
48 views

Proving that $\lim_{x \to\infty}\int_{x}^{x+1} f(t) dt=0$ if $\int_{1}^{\infty}f(x)<\infty$

I'm stuck with this problem: Prove that $\lim_{x \to\infty}\int_{x}^{x+1} f(t) dt=0$ if $\int_{1}^{\infty}f(x)<\infty.$ I've tried applying the Cauchy's condition, but I think that I can't do ...
5
votes
9answers
219 views

Finding $\int_{-\infty}^\infty \frac{x^2}{x^4+1}\;dx$

$$\int_{-\infty}^\infty \frac{x^2}{x^4+1}\;dx$$ I'm trying to understand trigonometric substitution better, because I never could get a good handle on it. All I know is that this integral is ...
0
votes
1answer
100 views

Integrate $\frac{1}{\sqrt{1-x^4}}$ from $0$ to $1$.

I'm having some difficulties doing this improper integral: $$\int_{0}^{1}\frac{1}{\sqrt{1-x^4}}\,dx$$ We have the conflictive point at $1$, but I don't know how to do this integral. I know how to ...
3
votes
3answers
58 views

Show that $\int_1^\infty \frac{\ln x}{\left(1+x^2\right)^\lambda}\mathrm dx$ is convergent only for $\lambda > \frac{1}{2}$

Show that the improper integral $$\int_1^\infty \frac{\ln x}{\left(1+x^2\right)^\lambda}\mathrm dx$$ is convergent only for $\lambda > \frac{1}{2}$. We will show that the sequence of integrals ...
0
votes
1answer
52 views

Convergence of $\int_0^{1/2}\frac{1}{x^\alpha \log x}dx$

To establish, for which values of real parameter $\alpha$, the integral $$\int_0^{1/2}\frac{1}{x^\alpha \log x}dx$$ exists finite. For me, this problem is very difficult. Any suggestions please?
4
votes
1answer
78 views

Intuition of $\int_{-\infty}^{\infty}$?

What's the intuition of the improper integral $$\int_{-\infty}^{\infty}$$ Is it really integral over the entire domain $\mathbb{R}$?
3
votes
2answers
79 views

Convergence of $\sum_{n=1}^{\infty}f(n)$ $\iff $ convergence of $\int_1^{\infty}$f(x)dx

f is a real valued $C^1$ function on [0,$\infty$]. Suppose that $\int_1^{\infty}|f'(x)|dx$ converges. Show that Convergence of $\sum_{n=1}^{\infty}f(n)$ $\iff $ convergence of $\int_1^{\infty}$f(x)dx. ...
0
votes
1answer
15 views

Equality between limit and integral whose integrand diverges at some point.[Edited]

Let $f:[0,1]\times[0,1]\to\mathbb{R}\cup\{\pm\infty\}$ be a function such that, for a given point $\hat{x}\in(0,1)$, $f$ is continuous in $[0,\hat{x})\times[0,\hat{x})$ and ...
1
vote
2answers
57 views

Prove: $\int_0^{\pi/2} \cos(\sec^{\;p} x) dx$ converges, $p > 0$.

Prove that $\int_0^{\pi/2} \cos(\sec^{\;p} x) dx$ converges for all $p>0$. I tried to do a change of variables through $u = \sec^{\;p} x$, but it is not sufficient to use the comparison test ...
2
votes
1answer
108 views

How do I evaluate $ \int_{0}^{\infty} \frac{1}{\sqrt{2 \pi s}} e^{-z^{2}/2s} \cdot \frac{1}{2}e^{-s/2} \, ds$?

How do you evaluate: $$\displaystyle \int_{0}^{\infty} \frac{1}{\sqrt{2 \pi s}} e^{-z^{2}/2s} \cdot \frac{1}{2}e^{-s/2} \, ds=?$$
2
votes
1answer
59 views

Proving an Integral Inequality using the Cauchy-Schwarz inequality

Assuming Cauchy Schwarz inquality as follows... $$\left|\int_a^b{f(x)g(x)dx} \right|\le \left(\int_a^b{|f(x)|^2}dx\right)^{1/2}\left(\int_a^b{|g(x)|^2}dx\right)^{1/2} $$ Where $g(x)=0$ and ...
0
votes
0answers
22 views

Dominated convergence theorem for improper integrals and related topics

Let $f:[0,1]\to\mathbb{R}$ be a continuous function and $x_{\varepsilon}\in(0,1)$ be points such that $x_{\varepsilon}\to\bar{x}$ for some $\bar{x}\in(0,1)$ as $\varepsilon\to0$. For $\delta>0$ ...
0
votes
0answers
44 views

Improper integral of an exponential with finite polynomial argument

I am interested in finding a general solution (or solutions to as many special cases as possible where $N > 2$) of the integrals $$I_{x\ \in\ (-\infty,\infty)}(\alpha_1,\dots,\alpha_N) := ...
1
vote
6answers
96 views

How to prove that this improper integral does not converge?

Prove that $$\int_1^\infty\frac{e^x}{x (e^x+1)}dx$$ does not converge. How can I do that? I thought about turning it into the form of $\int_b^\infty\frac{dx}{x^a}$, but I find no easy way to get rid ...