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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2answers
77 views

Integral of exponential product function

I want to know the value of this integral: $$\int_{0}^{\infty}e^{-u}e^{-u^{\alpha}x}\mathrm{d}u$$ where $\alpha>0$, $x>0$. Thank you.
3
votes
2answers
108 views

How to calculate the improper integral $\int_0^\infty\left(\frac{1}{\sqrt{x^2+4}}-\frac{P}{x+2}\right)dx$

This is the first time I've seen a problem like this. I have no idea what to do. Detailed guidance would be of great help. For which values of P does the integral converge? ...
10
votes
1answer
239 views

Evaluating $\int_0^\infty \frac {\cos {\pi x}} {e^{2\pi \sqrt x} - 1} \mathrm d x$

I am trying to show that$$\displaystyle \int_0^\infty \frac {\cos {\pi x}} {e^{2\pi \sqrt x} - 1} \mathrm d x = \dfrac {2 - \sqrt 2} {8}$$ I have verified this numerically on Mathematica. I have ...
1
vote
1answer
50 views

If $\int _1^{\infty }f\left(x\right)dx$ converges absolutely then $\int _1^{\infty }\sin \left(x\right)f\left(x\right)dx$ exists

I'm in need of some assistance with a homework question (I'm doing some calculus work by myself and have gotten stuck on this question): "Prove or give a counter-example of of the following ...
3
votes
1answer
99 views

Solve $\mathscr{F}^{-1} [ \cot{a \omega} \times \mathscr{F} \{ U(t) \sin{\omega_0 t} \} ] $ using contour integration

I wish to evaluate $y(t) = \mathscr{F}^{-1} [ \cot{a \omega} \times \mathscr{F} \{ U(t) \sin{\omega_0 t} \} ] $, where $\mathscr{F}$ represents the Fourier transform, and U(t) represents the ...
6
votes
5answers
270 views

How to find $ \int_0^\infty \dfrac x{1+e^x}\ dx$

How to find $$ \int_0^\infty \dfrac x{1+e^x}\ dx=\ ...? $$ I don't know where should I start with. The correct answer from my textbook is $\frac{\pi^2}{12}$. This is my homework with 10 questions ...
12
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6answers
267 views
1
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0answers
49 views

The Fourier sine transform of $f(x)/\sin x$

Is the following result $$\lim_{\lambda \to \infty} \frac{2}{\pi} \int_0^\infty \frac{f(x)}{\sin x} \sin(\lambda x) \, dx = f(0) + 2\sum_{k = 1}^\infty f(k\pi),$$ where $\lambda$ is an odd integer, ...
1
vote
0answers
95 views

Expectation of $\cos(\|X\|)$ where $X \sim \mathcal{N}(\mu,\Sigma)$

Do: $$ \int_{-\infty}^\infty \int_{-\infty}^\infty \cos\left(\sqrt{x^2+y^2}\right) e^{-\frac{1}{2}\left[\frac{(x-\mu_x)^2}{\sigma_x^2} + ...
10
votes
3answers
261 views

Definite integral - closed form: $\int_{0}^{\infty}\cos\left(x^{4} + 1 \over x^{2}\right)\,{\rm d}x$

I'm struggling with this definite integral: $$ \int_{0}^{\infty}\cos\left(x^{4} + 1 \over x^{2}\right)\,{\rm d}x. $$ Any help will be greatly appreciated.
3
votes
3answers
157 views

Does $\int _1^{\infty }\left(\sin \left(x^2\right)\right)dx$ converge or diverge? [duplicate]

I'm in need of some assistance regarding a question in my Calculus textboox: Find if the following converges or diverges without calculating the integral: $$\int _1^{\infty }\left(\sin ...
7
votes
2answers
119 views

if $\int_1^\infty f(x)dx$ exist, then $\int_1^\infty f^2(x)dx$ exist?

I'm facing some difficulty in proving/disproving this sentence: Consider $f: [0, \infty ) \rightarrow \mathbb{R}, f$ is continuous. if $\displaystyle \int_1^\infty f(x)dx$ exist, then ...
0
votes
2answers
56 views

How to determine if this integral converges? $ \int_{1}^{\infty} \frac{\cos x}{x}dx $

I should determine whether this is a convergent or divergent integral. I need to use the comparison test but I don't find any intgeral that helps me figure this. $$ \int_{0}^{\infty} \frac{\cos ...
5
votes
1answer
107 views

Solving the integral $\int_{0}^{\infty} \frac{\sin{x}}{x^2+1} dx$

I'm trying hard to solve this integral but I still don't know how... $$\int_{0}^{\infty} \frac{\sin{x}}{x^2+1} dx$$ The integral from $-\infty$ to $\infty$ is quite easy, but how could we integrate ...
3
votes
2answers
121 views

Integral of exponential using error function

I'm trying to solve some integrals below $$\int_{-\infty}^{\infty} {x^n e^\frac{-(x - \mu)^2}{\sigma^2}}dx$$ I am interested in the solutions where n = 0, 1, 2, 3, 4. I have learned that ...
1
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3answers
56 views

Let $(f_n)_{n\in\mathbb{N}} \rightarrow f$ on $[0,\infty)$. True or false: $\lim_{n\to\infty}\int_0^{\infty}f_n(x) \ dx = \int_0^{\infty}f(x) \ dx.$

The Assignment: Let $(f_n)_{n\in\mathbb{N}}$ converge uniformly to $f$ on $[0,\infty)$ and let the improper integrals of $f_n$ and $f$ exist on $[0,\infty)$. True or false: ...
3
votes
2answers
127 views

Evaluate the definite integral $ \int_{-\infty}^{\infty} \frac{\cos(x)}{x^4 +1} \ \ dx $

I am having more trouble with this problem then I feel like I should be. I set $ \ \cos(x) = e^{ix} \ $ and $ \ x^4 +1 = e^{\pi i /4} \ $ or $ \sqrt{i} \ $. I think I am suppose to do a residue to ...
8
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7answers
378 views

Calculus Question: $\int_0^{\frac{\pi}{2}}\tan (x)\log(\sin x)dx$

Can anyone help me to find $\int_0^{\frac{\pi}{2}}\tan (x)\log(\sin x)dx$? Any help would be appreciated. Thanks in advance.
2
votes
1answer
31 views

Is this proposition true or false?

Let $$f:[1,\infty)\to \mathbb R$$ such that $$\exists I_{n}=\int_{1}^{n}f(x)dx \forall n\in \mathbb N$$If $$\lim_{x\to \infty}f(x)=0; \lim_{n\to \infty}I_{n}=A$$ then $$\Rightarrow ...
1
vote
1answer
41 views

Complex Integral - exponential divided by a monomial

How does one solve integrals like this- $$I=\int^\beta_0 dx \frac{\exp(i\omega_nx)}{x-a}$$ where $\omega_n=\frac{\pi n}{\beta} $. EDIT: $\beta$ is a finite, real ...
2
votes
0answers
63 views

Multivariable Integral Calculus help

I have two questions. First: Is my proof "strong" enough? I am being asked to prove that $$\int_{0}^\infty\int_{0}^x e^{-sx}f(x-y,y) dydx = \int_{0}^\infty\int_{0}^\infty e^{-s(u+v)}f(u,v) dudv$$ ...
0
votes
1answer
27 views

how to find the right integral for comparison test?

I should determine whether this is a convergent or divergent integral. i need to use the comparison test but i don't know where to start. $$ \int_{1}^{\infty} e^{-\sqrt{x}}dx $$
0
votes
2answers
48 views

Show that Improper integral converges

Show that the following integral is convergent. $$ \int_{1}^{\infty}\sin\left(1 \over x^{2}\right)\cos\left(x^{2}\right)\,{\rm d}x $$ Not sure how I can solve ...
1
vote
1answer
20 views

how to use comparison test for improper integral?

I should determine whether this is a convergent or divergent integral. i need to use the comparison test but i don't know where to start. there is a method to find the integral we need to compare to? ...
1
vote
1answer
26 views

how to determine if this integral converge or not?

I should determine whether this is a convergent or divergent integral. The problem is that I don't know how to start. i need to use the comparison test but i don't know where to start. $$ ...
0
votes
1answer
21 views

Evaluating an integral arising from applying Fourier Transforms to a PDE

How an I evaluate the following integral: $$\frac{u_0}{\pi}\int_{-\infty}^{\infty}\frac{\sin( \alpha)\cos(\alpha x)}{\alpha}e^{-k \alpha^2 t}d \alpha \ \ ?$$ It arises as the solution $u(x,t)$ of the ...
0
votes
0answers
60 views

When does $\int_2^\infty \frac{1}{x^p \log^q x} dx$ converge?

What are the values of $p$, $q$ for which an improper below is convergent?: $$\int_2^\infty \frac{1}{x^p \log^q x}dx$$ I divided the case and did a comparison test, substitution. But I want an ...
2
votes
2answers
63 views

Convergence of $\int_{-\infty}^{\infty}f(x)dx$

I posed a question to my calculus professor, asking how to evaluate the Riemann integral $$\int_{-\infty}^\infty f(x) \, dx$$ I can simplify the above integral as $$\int_{-\infty}^{\infty}f(x)\,dx ...
0
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0answers
42 views

For what $p$, $\int_0^{\infty} \frac{\ln (1+x^{3p})}{(x+x^3)^{4p}arctg\sqrt{x}}$ converges

I have to see for what values of $p$ the following integral converges. $$\int_0^{\infty} \frac{\ln (1+x^{3p})}{(x+x^3)^{4p}arctg\sqrt{x}}=\int_0^1 \frac{\ln (1+x^{3p})}{(x+x^3)^{4p}arctg\sqrt{x}} + ...
7
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4answers
365 views

The closed form of $\int_0^{\infty} \frac{\log(\cosh(x))}{x} e^{-x} \ dx$

An integral I discussed last days in a chat, and it looks like a hard nut since after some manipulations of the initial form we reach an integral where the integrand is expressed in terms of ...
5
votes
1answer
84 views

Please help with integral

Please help me with evaluate the following improper integral $$\int_{0}^{\infty} \frac{\ln (1+u) -\ln 2}{(u+1)\sqrt{u} \ln u} du.$$
0
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0answers
26 views

Validity of $\int_{a}^b f(z) dz = \int_{a}^{\infty} f(z) dz - \int_{b}^{\infty} f(z) dz$?

What should be the conditions on the complex-valued function $f$ to be able to write : $$\int_{a}^b f(z) dz = \int_{a}^{\infty} f(z) dz - \int_{b}^{\infty} f(z) dz $$
1
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1answer
147 views

$\int_0^\infty x^{-\frac{3}{2}}e^{-\frac{(x-1)^2}{x}}dx=\int_0^\infty x^{-\frac{1}{2}}e^{-\frac{(x-1)^2}{x}}dx$

This is my third question following the previous post. Prove that \begin{equation} \int_0^\infty x^{-\large\frac{3}{2}}e^{-\large\frac{(x-1)^2}{x}}dx=\int_0^\infty ...
1
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2answers
59 views

a question about integral with parameter variables?

I have a problem proving $$\int_{0}^\infty dx {\left(\int_{0}^\infty e^{-x^2t}\sin t\, dt\right)}=\int_{0}^\infty dt\left( \int_{0}^\infty e^{-x^2t}\sin t\, dx\right)$$. I have been struggling for ...
5
votes
3answers
116 views

Evaluate Gauss-like Integral

Evaluate Integral $$\int_0^\infty e^{-ay^{2}-\frac{b}{y^2}}dy $$ Where a and b are real and positive. This integral is eerily similar to the Gaussian integral $$\int_0^\infty e^{-\alpha x^2}dx = ...
0
votes
1answer
24 views

Oscillatory integral and Van der Corput

I have questions about an oscillatory integral. Physics papers say the oscillations should "cancel each other out". By this logic, does this integral converge? $$ \int_0^{\infty} e^{-i x^3} \, dx ...
0
votes
0answers
23 views

Convergent Improper Integral help

I am currently studying improper integrals and came across the following problem. Analyze the convergence of the improper integral of $f(x,y) = 1 / ( x^4 + y ^2 ) $ over $R = \{(x,y) : x\geq 1, y\geq ...
4
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1answer
54 views

Does the integral $\int_0^1\frac{\sin x}{\sqrt{x^3}}\cos\left(\frac1x\right)dx$ converge?

I tried to prove convergence this way: Suppose: $f(x)=\left|\frac{\sin x\cos\left(\frac1x\right)}{\sqrt{x^3}}\right|$, $g(x)=\left|\frac{\cos\frac1x}{\sqrt x}\right|$. $\lim_{x\to ...
0
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1answer
70 views

a question about integral proof: $\lim_{n\to \infty} \int_{0}^\infty {n\cdot {\ln(1+{f(x)\over n}}})dx=\int_{0}^\infty f(x)dx$

A non-negative function ${\rm f}\left(x\right)$ is continuous in $(0,\infty)$ and $\displaystyle{\int_{0}^{\infty}{\rm f}\left(x\right)\,{\rm d}x}$ is convergent. Then, we need to prove $$\lim_{n\to ...
2
votes
4answers
122 views

$\int_{0}^{\infty}\frac{x}{x^3+1}dx$ =?

So guys I have this improper integral $\int_{0}^{\infty}\frac{x}{x^3+1}dx$. I checked that it converges by $ \int_{0}^{1}\frac{x}{x^3+1}dx + \int_{1}^{\infty}\frac{x}{x^3+1}dx $ and using the ...
1
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2answers
43 views

2d integral with difference of squares in exponent

$$\int_0^xdt\int_{-\infty}^tds\;e^\frac{t^2-s^2}{2}$$ Could anyone help me out with this integral? Polar coordinates helps but introduces difficulties with the boundaries, I'm not sure how to ...
0
votes
0answers
23 views

Estimates of an integral

I came across the following type of integral $$ I(r)=\int_r^{\infty}\frac{(\sinh s)^{1-\frac{n}{2}}}{\sqrt{\cosh s-\sinh r}}e^{is} ds $$ Can I have the bound $I(r)\leq C r^{-\frac{n-1}{2}}$ as $r\to0 ...
0
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2answers
561 views

Practical use and applications of improper integrals

What are the most important applications of improper integrals, in particular to computer science and related fields, and to technology and engineering in general? I know that improper integrals are ...
1
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1answer
36 views

A question about convergence of improper parametric integral

Could you give me some hint how to find all $\alpha\in R$ for with the integral $\int_0^1 \frac{a-x^{\alpha}}{1-x}$ converges. Is clear that this integral converges for all$\alpha\in N$, but I could ...
0
votes
1answer
31 views

How to compute this integration?

Afternoon, I am keeping in studying on exam and stumbled upon this integral (I am asked to count it with per-parted procedure) - $\int {2x}\arctan x\,dx$ How should I proceed the "$\arctan x$" ...
6
votes
1answer
74 views

Tricky Integral equation - where to start?

How would you go about solving this? $$p(x,t)=C\exp\left[-x+\int_0^t\int_0^\infty y\,p(y,\tau)\,\mathrm{d}y\,\mathrm{d}\tau\right]$$ Here $p(x,t)$ is the time-dependent probability distribution of a ...
4
votes
2answers
213 views

Definite integral $\int_{R_0}^{R}\frac{dr}{r^2\sqrt{\frac{R_0-R_S}{R_0^3}-\frac{1}{r^2}\left(1-\frac{R_{s}}{r}\right)}}$

In general relativity, null geodesics (in the unbounded case) can be written under the following form : ...
2
votes
2answers
113 views

Integration problem: $\int _ {-\infty} ^ {\infty} \frac {e^{-x^2}}{\sqrt{\pi}} e^x\ dx$

I have to integrate $$\int _ {-\infty} ^ {\infty} \frac {e^{\large-x^2}}{\sqrt{\pi}} e^{\large x}\ dx.$$ I've already done by numerical approximations, like Simpson's rule and Gauss-Hermite, but I ...
9
votes
6answers
453 views

Hints on calculating the integral $\int_0^1\frac{x^{19}-1}{\ln x}\,dx$

I would be happy to get some hints on the following integral: $$ \int_0^1\frac{x^{19}-1}{\ln x}\,dx $$
1
vote
2answers
47 views

Discuss the convergence of $\int_0^1x^n \left[\log({1\over x})\right]^m \, dx$

Discuss the convergence of $$\int_0^1x^n\left[\log\left({1\over x}\right)\right]^m \, dx$$ Need some clues. I know that both $0$ and $1$ are points of discontinuities.