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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4
votes
5answers
100 views

How to integrate $\int_{0}^{1}\ln\left(\, x\,\right)\,{\rm d}x$?

I encountered this integral in the quantum field theory calculation. Can I do this: $$ \left. \int_{0}^{1}\ln\left(\, x\,\right)\,{\rm d}x =x\ln\left(\, x\,\right)\right\vert_{0}^{1} ...
2
votes
1answer
49 views

Prove or disprove following integral.

Assume $L$ a constant, and assume $x$ real. Is the following equation true? $$ \int_{-\infty}^\infty\frac{1}{k^2}\exp(-ikx)dk = \frac{L}{|x|} $$ If it is true, find the value of $L$. If it is not, ...
0
votes
0answers
26 views

Banach Spaces: Improper Riemann Integral

Reference This thread is related to: Stone's Theorem For a bounded example of non-integrability see: Riemann Integral: Nonexample? For a comparison of integrals see: Uniform Integral vs. Riemann ...
1
vote
1answer
25 views

Cauchy Principal Value Problem: Gaussian and exponential over a quadratic

I need help with the following integral: $$ \int_{-\infty}^{\infty}\frac{e^{-x^2}e^{iax}}{1-x^2}dx$$ Where $a$ is real. Obviously the integral doesn't converge due to the singularities at $|x|=1$ ...
1
vote
1answer
34 views

I suspect this integral has a closed form but I can't find it

$$\int_{-\infty}^\infty \!\!\text{d} r\dfrac{1}{r}e^{\frac{-(r-r_0)^2}{\delta^2}}\sin(k r)$$ Where $\delta>0$, $r_0\in \mathbb{R}$. Can anyone help me with this? it seems to me there has to be a ...
3
votes
2answers
50 views

Find the Fourier transform of $u(x) = \frac{x \cos(2x)}{(1+x^2)^2}$

Find the Fourier transform of $$u(x) = \frac{x \cos(2x)}{(1+x^2)^2}$$ My work Okay so we want $$\int_\mathbb R \frac{e^{-ixt}x\cos(2x)}{(1+x^2)^2}dx$$ Of course we want to apply the residue ...
4
votes
7answers
166 views

Show that $\int_0^\infty \frac{\sin (\lambda x)}{e^x} \, \mathrm dx =\frac{\lambda}{1+{\lambda^2}}$

$$\int_0^\infty \frac{\sin (\lambda x)}{e^x} \, \mathrm dx =\frac{\lambda}{1+{\lambda^2}}$$ My intuition telling me there might be an $\arctan$ coming up, but I don't know how to do this ...
0
votes
2answers
56 views

Integral of 1/sinx between 0 and 1 diverges.

I am learning about ways to test if an integral converges or diverges and I am stuck with this one: $\displaystyle{\int{{\rm d}x \over \sin\left(\, x\right)}}$ between $0$ and $1$. The tests I know ...
2
votes
2answers
41 views

Convergence of $\int_0^\infty {\frac{\sin(x)(x+4)}{\sqrt{x^3(x+1)^2}}}$.

I am trying to check the convergence of $$ \int_0^\infty {\frac{\sin(x)(x+4)}{\sqrt{x^3(x+1)^2}}}\,dx. $$ I divided it into two cases, from 0 to 1 and from 1 to $\infty$. I could see, using modulus ...
0
votes
3answers
21 views

Choosing which function to compare to for the Direct Comparison test

$$\int_1^\infty (e^{-x^{2}})dx$$ why use $$e^{-x}$$ for the direct comparison test to determine convergence or divergence?
1
vote
1answer
19 views

Steps in evaluating infinite integral

This is my teacher's work. " " How does the the $a^2\ln(a)/2)$ lose the $1/2$ part when its limit is taken? I.E. the step following $a^2\ln(a)/2$ is limit as a approaches 0 from the right side of ...
0
votes
1answer
30 views

Divergence and Convergence of improper integrals of $1/x$ and $1/x^2$

Prove that $\int_1^\infty dx/x $ diverges and $\int_1^\infty dx/x^{2} $ converges I think that the former, $dx/x$ converges as plugging the bounds doesn't yield a non-existent result.
1
vote
1answer
26 views

Improper double integral

Can I apply the Fundamental Theorem of Calculus for $$\int_{-\infty}^{t_1} \int_{-\infty}^{t_2} \frac{\partial \phi\left(\frac{z_2 - \rho z_1}{\sqrt{1 - \rho^2}}\right)}{\partial z_2} dz_2 dz_1$$ in ...
3
votes
0answers
21 views

Local behavior of a Fourier series and a intgral

So I have to calculate an integral that involves a Fourier series of some function. I would like to get some kind of local control of the function near zero the series is ...
3
votes
2answers
55 views

Improper Integral of $\int\frac{dx}{(2x-1)^3}$

Improper Integral of $$\int_{-\infty}^0\frac{dx}{(2x-1)^3}$$ from Anton Calculus 8th Edition, page 576, question 9. Answer is $-\frac{1}{4}$ but I'm finding $-1$ The integral, substituting ...
7
votes
0answers
120 views

Evaluating $\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}dx$

Evaluate $$\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}dx$$ I tried using by parts and complex numbers along with series expansion but I was unable to find the answer. Please Help!
1
vote
1answer
40 views

Solving for C when we have $C\int_0^\infty \int_0^\infty \frac{e^\frac{-(x_1+x_2)}{2}}{x_1+x_2} \,dx_1 \,dx_2=1$

Solving for C when we have $C\int_0^\infty \int_0^\infty \frac{e^\frac{-(x_1+x_2)}{2}}{x_1+x_2} \,dx_1 \,dx_2=1$ $$\int_0^\infty \int_0^\infty \frac{e^\frac{-(x_1+x_2)}{2}}{x_1+x_2} \,dx_1 \,dx_2$$ ...
1
vote
2answers
24 views

Improper integral of $\int\frac{2}{x^2-1}$

Improper integral of $\int^\infty_3\frac{2}{x^2-1}dx $ I know I need the limit of $\lim_{b \to \infty}$. Solving the integral first: $$\int\frac{2}{x^2-1}dx = 2 \int\frac{1}{x^2-1}dx = 2\ln|x^2-1|$$ ...
14
votes
6answers
292 views

Evaluate $\int_0^1 \frac{x^k-1}{\ln x}dx $ using high school techniques

Is there a way to compute this integral, $$\int_0^1 \frac{x^k-1}{\ln x}dx =\ln({k+1})$$ without using the derivation under the integral sign nor transforming it to a double integral and then ...
2
votes
2answers
80 views

Closed form for $\int_0^\infty e^{-x}\sin^a(x)dx$

Can we find a closed form for $$I(a)=\int_0^\infty e^{-x}\sin^a(x)dx$$ Mathematica can easily find closed form for integer $a$: \begin{align*} I(0)&=1\\ I(1)&=1/2\\ I(2)&=2/5\\ ...
3
votes
1answer
46 views

What am I doing with this triple integral?

I am new here and hope my question is clear and is straight to the point. The following is a form of an integral I am trying to compute. $$\int_{x}\int_{y}\int_{z} f(x,y,z) g(x,y)\ dz \ dy \ dx \ ...
4
votes
1answer
71 views

Evaluating $\int_0^{\infty} \frac{\sqrt{x}}{x^2+2x+5} dx$ using complex analysis

how do I compute $$\int_0^{\infty} \frac{\sqrt{x}}{x^2+2x+5} dx$$ with complex analysis? I feel like im calculating the residue wrong and I cant get to the answer correctly. I tried to branch cut ...
2
votes
2answers
43 views

How to evaluate if $\int_2^\infty {\frac{1}{\log(x)\cdot \sqrt{x^2+1}}}dx$ converges?

I am asked to evaluate if $$\int_2^\infty {\frac{1}{\log(x)\cdot \sqrt{x^2+1}}}dx$$ converges. How can that be done? Even Wolframalpha/Mathematica 8.0 does not return a value. Can this be done with ...
0
votes
1answer
55 views

Integration of $\exp[f(x,y)]$

Here is the question i want to solve. $$\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \exp\left[{-2\over3}(y^2-yz+z^2)\right]\,dy\,dz$$ I know that $\exp$ is $e^{f(x)}$ and i can find $\int ...
2
votes
1answer
39 views

Limit of an integral that arose in Fourier Analysis

$$\lim_{\alpha\to\infty} \int_{0}^{\delta}\frac{\sin(\alpha x) \sin(\lambda\alpha x)}{x^2}dx$$ where $\lambda,\delta \in \mathbb{R}, \lambda,\delta > 0$. Appreciate your help in finding this limit. ...
1
vote
3answers
234 views

Example of a function $f(x)$ such that the integral of $f(x^2)$ converges but the integral of $f(x)$ diverges?

Does anybody know an example of a function $f(x)$ such that the integral from $1$ to infinity of $f(x^2)$ converges but the integral of $f(x)$ from $1$ to infinity diverges? Thanks!
0
votes
0answers
13 views

Evatuating integral $\int_{-\infty}^{+\infty} 2\chi^2(\rho+v(t - \tau))(V_1 + V_2 \sin(\omega t))e^{-\chi^2(\rho+v(t - \tau))^2}dt$

I am taking analytical mechanics course, I need to solve integral $$\int_{-\infty}^{+\infty} 2\chi^2(\rho+v(t - \tau))(V_1 + V_2 \sin(\omega t))e^{-\chi^2(\rho+v(t - \tau))^2}dt$$ The textbook says ...
1
vote
0answers
40 views

Trying to evaluate integral using complex analysis

Again, improper integrals involving $\ln(1+x^2)$ I am trying to get a result for the integral $I_{\alpha}=\int_{0}^{\infty} \frac{\ln(1+x^2)}{x^\alpha}dx$ - asked above link- using some complex ...
0
votes
0answers
21 views

Evaluating integral $\int_{-\infty}^{+\infty} 2\chi^2(\rho+v(t - \tau))(V_1 + V_2 \sin(\omega t))e^{-\chi^2(\rho+v(t - \tau))^2}dt$

I am taking analytical mechanics course, I need to solve integral $$\int_{-\infty}^{+\infty} 2\chi^2(\rho+v(t - \tau))(V_1 + V_2 \sin(\omega t))e^{-\chi^2(\rho+v(t - \tau))^2}dt$$ The textbook says ...
1
vote
1answer
31 views

Why does an improper integral turn into an answer with factorial?

Suppose I have $\int_{0}^{\infty}y^{2n+1}e^{-y}dy$ Why does this integral equal $(2n+1)!$ ? Could somebody please explain this?
0
votes
2answers
17 views

Behavior of Improper Integral

I am trying to understand better the behavior of improper integrals depending on the function. I think that this items are correct by intuition, but I can't seem to find a theorem or lemma that ...
0
votes
1answer
24 views

How do I manipulate orders of integration to cancel out a nested integral leaving me with only 1?

Suppose $A$ and $B$ are random variables and the only restriction on the probability density function $f_{A,B}(a,b)$ is $P(B>A) = 1$. Also suppose $p(x) = P(A \leq x \leq B)$ where $x$ is $fixed$ ...
0
votes
1answer
54 views

The limit of $ m\int_{a}^{1/m} \frac{dx}{x}=0 $ and $ m\int_{a}^{\infty} \frac{dx}{x^{1+m}}=0$ as $m\to0$

Given $ a >0 $ is it correct that $$ \lim_{m\to 0}m\int_{a}^{1/m} \frac{dx}{x}=0 $$ by the properties of the logarithm function? Or on the other hand, $$\lim_{m\to 0} m\int_{a}^{\infty} ...
6
votes
2answers
107 views

Evaluate $\int_0^\infty \frac{\arctan(3x) - \arctan(9x)}{x} {dx}$

Evaluate the integral $$\int_0^\infty \frac{\arctan(3x) - \arctan(9x)}{x} {dx}.$$ I tried to split this into 2 integrals and then using the substitution $t = \arctan(3x)$ but I got nowhere.
0
votes
1answer
29 views

Convergence test for improper multiple integral

I have a function $f:\mathbb R^n \to \mathbb R$ such that $f(x)=(1+|x|)^me^{-\frac{|x|^2}{a}}$. I need to check is $$\int\limits_{\mathbb R^n}f(x)dx = \int\limits_{\mathbb R^n} ...
1
vote
2answers
61 views

Is $\int_0^{\pi/2} \sin^a(\theta)\tan(\theta) d\theta $ convergent?

Could someone please help me with this question? I'm not sure how i should manipulate $\sin\theta$ because of $a$. Determine if the improper integral converges: $\int_0^{\pi/2} ...
0
votes
1answer
34 views

Convergence of $ \int_2^\infty \frac{1}{x^q(\ln x)^p} dx $

Could someone help me out with this question? I'm studying for my exam and I'm stuck. Question: Let $p,q \in (-\infty, \infty).$ Find all $(p,q)$ such that the improper integral $ \int_2^\infty ...
0
votes
1answer
33 views

Orthogonality of Hermite functions

I would like to prove to myself that Hermite functions, defined by $\varphi_n(x)=(-1)^n e^{x^2/2}\frac{d^n e^{-x^2}}{dx^n}$, $n\in\mathbb{N}$ are an orthogonal system in $L^2(\mathbb{R})$, i.e. that, ...
0
votes
1answer
34 views

Divergence of double integral $\iint_{x^2 + y^2 \leq 1} \frac{\mathop{d}x\mathop{d}y}{(x^2 + xy + y^2)^p},\quad p \geq 1$

How to prove the divergence of the integral $$ \iint_{x^2 + y^2 \leq 1} \frac{\mathop{d}x\mathop{d}y}{(x^2 + xy + y^2)^p},\quad p \geq 1 $$ without transition to generic polar coordinate system?
7
votes
4answers
115 views

show $\int^\infty_0 e^{-sx} x^{-1} \sin{x} dx = \frac14 \log{(1+4s^{-2})}$ for $s>0$

This is problem 2.6.58 of Folland's Real Analysis book: show $\int^\infty_0 e^{-sx} x^{-1} \sin{x} dx = \frac14 \log{(1+4s^{-2})}$ for $s>0$ by integrating $e^{-sx} \sin{(2xy)}$ over x and y. I ...
2
votes
0answers
15 views

Can this divergent integral transform be regularized?

The integral $$\int_0^{\infty} e^u \ K_{i t}(u) du$$ is the adjoint Kontorovich-Lebedev transform of the increasing exponential function, but unfortunately this integral is divergent because $$e^u \ ...
0
votes
0answers
33 views

How to do contour integration? [duplicate]

I've searched a lot throughout the web, but havent found anything yet, so I am posting my own question. I am very interested in complex analysis, hopefully someone can help me out here. Suppose we ...
2
votes
1answer
45 views

Determine how large the number a has to be?

This is what i've done so far: i converted to limit notation lim as t goes to infinity of integral from a to t of 1/t^2+1 dt lim as to goes to infinity [arctan(t)] from a to t (lim as t goes to ...
0
votes
1answer
38 views

Integral of $\int_0^{\infty} x^{4n+3} e^{-x} \sin x dx$.

Can some one help me with the integral $$\int_0^{\infty} x^{4n+3} e^{-x} \sin x dx$$ According to my exercise I should be able to get $0$. Please help me .
10
votes
3answers
308 views

Closed form of $\int_{0}^{\infty} \frac{\tanh(x)\,\tanh(2x)}{x^2}\;dx$

I have homework to evaluate this integral $$I=\int_{0}^{\infty} \frac{\tanh(x)\,\tanh(2x)}{x^2}\;dx$$ Here is what I have done so far. I tried integration by parts using $u=\tanh(x)\,\tanh(2x)$ and ...
1
vote
3answers
71 views

Does $\sum \frac{(n+4^n)}{n+6^n}$ converge or diverge?

The Question Does $\sum \frac{(n+4^n)}{n+6^n}$ converge or diverge? Please note I have no knowledge of Alternating Series, Ratio and Root tests, Power Series, or Taylor and McLaurin Series. My Work ...
6
votes
3answers
103 views

How to evaluate this improper integral?

I got stuck when evaluating these two improper integrals:$$ \int_a^b\frac{dx}{\sqrt{(b-x)(x-a)}} $$ and$$ \int_0^1\frac{dx}{\sqrt{x-x^3}} $$ How to evaluate them? Thank you!
28
votes
2answers
423 views

A strange integral

While browsing on Integral and Series, I found a strange integral posted by @sos440. His post doesn't have a response for more than a month, so I decide to post it here. I hope he doesn't mind because ...
-2
votes
0answers
73 views

How to evaluate $\int_0^1\frac{\ln^2(x)\ln(1+x^2)}{1-x}dx$ and$\int_0^1\frac{\ln^2(x)\ln(1+x^2)}{1+x}dx$ [on hold]

How to express these integrals How to evaluate$$ \int_0^1\frac{\ln^2(x)\ln(1+x^2)}{1-x}dx$$ and$$\int_0^1\frac{\ln^2(x)\ln(1+x^2)}{1+x}dx$$
1
vote
1answer
28 views

How do you evaluate an exponential term that contains both $-\infty$ and $+\infty$?

What does $\int_{0}^{\infty} e^{y(iu-\alpha)}dy = ?$ Please note $i$ is a complex variable, $\alpha$ and $u $ are constants. I know this integral evaluates to: ...