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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2
votes
1answer
34 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
5answers
94 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} ...
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 ...
41
votes
5answers
2k views

Calculating the integral $\int_{0}^{\infty} \frac{\cos x}{1+x^2}\mathrm{d}x$ without using complex analysis

Suppose that we do not know anything about the complex analysis (numbers). In this case, how to calculate the following integral in closed form? $$\int_0^\infty\frac{\cos\;x}{1+x^2}\mathrm{d}x$$
6
votes
6answers
279 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.
9
votes
4answers
2k views

Improper Integral Question: $ \int_0 ^ \infty\frac{x\log x}{(1+x^2)^2} dx $

I have to test the convergence of the integral : $$ \int_0 ^ \infty\frac{x\log x}{(1+x^2)^2} dx $$ Please suggest. Also, have to show that the value of the integral is zero ?
2
votes
1answer
48 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, ...
16
votes
6answers
566 views

Closed form for $\int_{0}^{1/2}\left(2x - 1\right)^{6}\ \log^{2}\left(2\sin\left(\pi x\right)\right)\,{\rm d}x$

How can I find a closed form for the following integral $$ \int_0^{1/2}\left(2x - 1\right)^{6}\ \log^{2}\left(2\sin\left(\pi x\right)\right) \,{\rm d}x $$
0
votes
0answers
23 views

Banach Spaces: Improper Riemann Integral

Disclaimer This thread is related to: Stone's Theorem Definition Given a measure space $\Omega$ and a Banach space $E$. Consider functions $F:\Omega\to E$. Denote the measurable subsets of finite ...
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 $|x|=1$ ...
23
votes
3answers
554 views
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 ...
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 ...
7
votes
3answers
120 views

Prove that $\int_0^\infty \frac{e^{\cos(ax)}\cos\left(\sin (ax)+bx\right)}{c^2+x^2}dx =\frac{\pi}{2c}\exp\left(e^{-ac}-bc\right)$

In my course, I have to prove formula below $$I=\int_0^\infty \frac{e^{\cos(ax)}\cos\left(\sin (ax)+bx\right)}{c^2+x^2}dx =\frac{\pi}{2c}\exp\left(e^{-ac}-bc\right)$$ for $a,b,c>0.$ I know that ...
14
votes
6answers
469 views

Proving $\displaystyle\int_0^1\frac{\ln(x)}{x^2-1}\,\mathrm dx=\frac{\pi^2}{8}$

How can I prove that? $$\int_0^1\frac{\ln(x)}{x^2-1}\,\mathrm dx=\frac{\pi^2}{8}$$ I know that $$\int_0^1\frac{\ln(x)}{x^2-1}\,\mathrm ...
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} ...
10
votes
2answers
177 views

Elegant proof of $\int_{-\infty}^{\infty} \frac{dx}{x^4 + a x^2 + b ^2} =\frac {\pi} {b \sqrt{2b+a}}$?

Let $a, b > 0$ satisfy $a^2-4b^2 \geq 0$. Then: $$\int_{-\infty}^{\infty} \frac{dx}{x^4 + a x^2 + b ^2} =\frac {\pi} {b \sqrt{2b+a}}$$ One way to calculate this is by computing the residues at the ...
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 ...
14
votes
6answers
291 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 ...
7
votes
3answers
227 views

Using differentiation under integral sign to calculate a definite integral

I want to calculate the integral $$\int^{\pi/2}_0\frac{\log(1+\sin\phi)}{\sin\phi}d\phi$$ using differentiation with respect to parameter in the integral ...
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 ...
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!
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 ...
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
1answer
277 views
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: $$\int\frac{2}{x^2-1}dx = 2 \int\frac{1}{x^2-1}dx = 2\ln|x^2-1|$$ ...
0
votes
1answer
324 views

Improper integral of a function involving square root and absolute value.

$$\int_{-2}^{8}\dfrac{dx}{\sqrt{|2x\|}}$$ I understand that you have to split this into two integrals because at $x=0$, the function is not defined. The example showed that they split up the integral ...
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
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 \ ...
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 ...
15
votes
1answer
580 views

Closed form for this integral $\int_{0}^{\infty}\frac{dx}{\sqrt{x}}\, e^{-x^{2}-\frac{b^{2}}{x}}$

How would you evaluate this integral? \begin{equation}\int_{0}^{\infty}\frac{dx}{\sqrt{x}}\, e^{-x^{2}-\frac{b^{2}}{x}}\end{equation} It reminds me of the form of a modified Bessel function of the ...
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 ...
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
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!
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. ...
7
votes
4answers
218 views

How to evaluate $\int_0^\infty \frac{1}{x^n+1} dx$

Noticed that the integral $$\int_0^\infty \frac{1}{x^n+1} dx$$ is often approached with partial fraction decomposition, but this gets increasingly ugly as $n$ gets bigger. Is there a neat trick to do ...
26
votes
4answers
800 views

Prove $\int_0^\infty \frac{\ln \tan^2 (ax)}{1+x^2}\,dx = \pi\ln \tanh(a)$

$$ \mbox{How would I prove}\quad \int_{0}^{\infty} {\ln\left(\,\tan^{2}\left(\, ax\,\right)\,\right) \over 1 + x^{2}}\,{\rm d}x =\pi \ln\left(\,\tanh\left(\,\left\vert\, ...
6
votes
4answers
173 views

Again, improper integrals involving $\ln(1+x^2)$

How can I check for which values of $\alpha $ this integral $$\int_{0}^{\infty} \frac{\ln(1+x^2)}{x^\alpha}\,dx $$ converges? I managed to do this in $0$ because I know $\ln(1+x)\sim x$ near 0. I ...
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?
5
votes
6answers
136 views

Find $\int_{ - \infty }^{ + \infty } {\frac{1} {1 + {x^4}}} \;{\mathrm{d}}x$

How can we find the integral: $$\int_{ - \infty }^{ + \infty } {\frac{1} {1 + {x^4}}} \;{\mathrm{d}}x$$ I tried to find and got it to be $\cfrac{\pi}{\sqrt2}$. Am I correct? Please help me with an ...
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 ...
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: ...