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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1answer
10 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? ...
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1answer
25 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. $$ ...
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1answer
20 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 ...
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0answers
54 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 ...
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2answers
50 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 ...
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0answers
37 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}} + ...
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4answers
262 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
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1answer
74 views

Please help with integral

Please help me with evaluate the following improper integral $$\int_{0}^{\infty} \frac{\ln (1+u) -\ln 2}{(u+1)\sqrt{u} \ln u} du.$$
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0answers
25 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 : $$\int_{a}^b f(z) dz = \int_{a}^{\infty} f(z) dz - \int_{b}^{\infty} f(z) dz $$
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1answer
131 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 ...
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2answers
54 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 ...
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3answers
80 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 = ...
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1answer
14 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 ...
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0answers
20 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 ...
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1answer
43 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 ...
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1answer
63 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
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4answers
117 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 ...
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2answers
41 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 ...
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0answers
21 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 ...
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2answers
81 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 ...
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1answer
23 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 ...
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1answer
30 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 "$\arctan x$" ...
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1answer
67 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
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2answers
158 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
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2answers
98 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 ...
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4answers
361 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: $$ \int_0^1\frac{x^{19}-1}{\ln x}\,dx $$
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2answers
40 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.
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2answers
39 views

Prove that $\int_{0}^{\infty}\left|e^{-xt}\frac{1-\cos(x)}{x}\right|dx$ converges

As part of one calculation I want to show that the following integral converges absolutely: $$\int_{0}^{\infty}e^{-x}\frac{1-\cos(x)}{x}dx$$ wihtout calculating its value. Using integral handbooks ...
3
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2answers
64 views

How to prove divergence of the integral $\int_{0}^{\infty}\frac{\sin(x)}{x^{2}}dx$

I want to show that the following integral diverges: $$\int_{0}^{\infty}\frac{\sin(x)}{x^{2}}dx$$ I used the substitution $ t = \frac{1}{x} $ to transform this integral into $$\int_{0}^{\infty}\sin ...
5
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1answer
61 views

Lobachevsky's Formula for Integrals

If the function $f:\mathbb{R}\to \mathbb{R}$ satisfies $f(x+\pi)=f(\pi-x)=f(x), \forall x \in \mathbb{R}$ then $$\int_0^\infty f(x)\frac{\sin x }{x} \mathrm{d}x = \int _0^\frac\pi2 f(x) \mathrm{d}x$$ ...
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3answers
56 views

Determining the convergence of $\int_{1}^{2}\frac{1}{\sqrt{x}(2-x)}\, \mathrm{d}x$ in simple way.

Is there a (simple) way to determine its convergence without determining its value? I know that the function $x\mapsto \left (\sqrt{x}(2-x) \right )^{-1}$ is continuous for all $x\in ...
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1answer
18 views

Convergence of improper Integral $\lim_{x -> 0} x^{\mu+n}{{\log x}\over {1+x}}$

$$\int_0^1{{x^n\log x}\over {(1+x)^2}} $$ My Attempt : The integrand is unbounded for $x=0$. so, Using $\mu-Test$ , the integral will be convergent if $0<\mu<1$ such than$$\lim_{x -> 0} ...
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1answer
68 views

Show that the improper integral $\int_1^\infty f(x) \ dx$ exists iff $\sum_1^\infty a_n$ converges.

The assignment: Let $(a_n)_{n\in\mathbb{N}}$ be a sequence of real numbers and $f: [1, \infty) \rightarrow \mathbb{R}$ be a function, defined by $f(x) = a_n$, for $x \in [n,n+1).$ Show that: ...
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0answers
78 views

About differentiation under the integral sign

I would like to ask something related to the application of the differentiation under the integral sign (Leibniz rule) given by ...
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1answer
41 views

Evaluate the integral using principal value and complex analysis

I need to find the value of the integral: $\int_{-\infty}^{\infty} \frac{sin^2x}{x^2}dx $ Right now progress: Because the value of $\frac{sin^2x}{x^2}$ is convergent, the integral will be equal ...
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2answers
53 views

Improper Integral [closed]

I manage to do improper integral however I just would like to check if the answer I worked out is correct or wrong: $$ \begin{array}{l} (1)\qquad\int\limits_{-\infty}^\infty \frac{dx}{729+x^6} = ...
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1answer
125 views

Evaluation of definite integral using complex analysis

I want to evaluate the following indefinite integral $$ \int_0^{\infty} x^{p - 1} \cos (ax) dx$$ where $0 < p < 1$ and $a > 0$. I was considering the function $f(z) = z^{p - 1} e^{iaz}$ and ...
3
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1answer
54 views

When $\int_{0}^{\infty}f(x)dx=\sum_{n=0}^{\infty}\int_{n}^{n+1}f(x)dx$?

Is the following always true? (i.e. if both converges, limits are equal; if one diverges, the other must diverge; EXCLUDE the case where the limit keeps "jumping") $$ ...
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0answers
44 views

Bound on Bessel potential

Let $s\in\mathbb{C}$. For a complex number $z$, $Re(z)>0$, consider the Bessel potential $$K_s(z)=\int_0^{+\infty}e^{-z\cosh t}\cosh(st)dt$$ I need to prove that, if $|z|\leq 1$, then ...
5
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1answer
127 views

Log Sine: $\int_0^\pi \theta^2 \ln^2\big(2\sin\frac{\theta}{2}\big)d \theta.$

Hi I am trying to calculate $$ I:=\int_0^\pi \theta^2 \ln^2\big(2\sin\frac{\theta}{2}\big)d \theta. $$ Here is a related Integral...$\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d ...
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1answer
58 views

$\int_{0}^{\infty} \frac{1 - e^{-px} \sin(x)}{x} dx$ Evaluate Integral

How do I solve the integral: $$\int_{0}^{\infty} \frac{1 - e^{-px} \sin(x)}{x} dx$$ I know the answer is $ \arctan(p) $ but have no idea as to how to show that. Any hints welcome! Thanks
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0answers
33 views

Question about convergence of improper integral

Could you give me some hint how to solve this problem: Suppose $f$ is continuous on $(0,1]$ and there is $M$ such as $\left|\int_x^1f(t)\, dt \right|\le M$. Prove that $\int_0^1f(x)\, dx$ converges ...
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2answers
54 views

Compute an improper integral.

Suppose $A = [0,\infty) \times [0, \infty) $. Let $f(x,y) = (x+y)e^{-x-y} $. How can I find $ \int_A f $? I know since $f$ is continuous on $A$, then $\int _A f $ exists, Do I need to evaluate $$ ...
5
votes
5answers
207 views

Evaluating the integral $\int_0^\infty \frac{x \sin rx }{a^2+x^2} dx$ using only real analysis

Calculate the integral$$ \int_0^\infty \frac{x \sin rx }{a^2+x^2} dx=\frac{1}{2}\int_{-\infty}^\infty \frac{x \sin rx }{a^2+x^2} dx,\quad a,r \in \mathbb{R}. $$ Edit: I was able to solve the integral ...
1
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1answer
42 views

Integration involving complicated exponential form

I'm trying to simplify the following: $\int_0^ts^{-\frac{3}{2}}e^{-\frac{(a+bs)^2}{2s}}~ds$ Basic substitution always gives a $s^{-\frac{1}{2}}~ds$ counterpart which I don't know how to get rid of. ...
1
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0answers
56 views

How to prove convergence of $\int_0^1f\left(\sqrt x \right)dx$?

Could you please give me some hint how to prove convergence of $\int_0^1f\left(\sqrt x \right)dx$ when f(x) is continuous for $0<x\le1$ and $\lim_{x\to0^+}x^3f^2(x)=1$ ? I tried the usual way: ...
2
votes
1answer
91 views

How to integrate $\displaystyle\int_o^\pi\frac{dx}{\sqrt{3-\cos(x)}}$?

How to integrate $\displaystyle\int_o^\pi\frac{dx}{\sqrt{3-\cos(x)}}$ ? If I take $y=\sin\left(\frac{x}{2}\right)$ then, $\displaystyle ...
1
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0answers
33 views

Convolving two functions

I'm trying to convolve two functions $f$ and $g$. $$f(x) = e^{-\frac{{(x-p_2)}^2}{2 q_2^2}}$$ $$g(x) = \left(i_1 e^{-\frac{(a-x)^2}{2 \sigma ^2}}+j_1 e^{-\frac{(b-x)^2}{2 \sigma ^2}}\right) \left(i_0 ...
0
votes
1answer
47 views

A question about improper integral

Could you please give me some hint how to solve this problem: Suppose f(x) continuous in $[0,\infty)$ and for each a,b>0 and c>b $ab \left|\int_0^1 f\left(ax+c \right) dx \right|<1$. Prove ...
4
votes
1answer
112 views

Show $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{1}{1+y^2}e^{-ay} dy =0 $

Need to prove $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{1}{1+y^2}e^{-ay} dy =0 $ and $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{y}{1+y^2}e^{-ay} dy =0 $ Can ...