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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0
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1answer
25 views

Convergence of integral of multiplication of two positive functions

I have two functions $f, g:\mathbb{R}\rightarrow \mathbb{R}_{\ge0}$, that are continuous. I know that $\int\limits_{-\infty}^\infty f(s) \, ds=C_1<\infty$, and $g(s)\le C_2$, with $C_1> 0$ and ...
2
votes
4answers
52 views

Prove that $\int_{-1}^{1} 4x \sqrt{1-x^{2}}\, dx = -3\pi$

What is of interest is the assertion $$\int_{-1}^{1} 4x \sqrt{1-x^{2}}\,dx = -3\pi.$$ Since $$\pi = 2\int_{-1}^{1}\sqrt{1-x^{2}}\,dx,$$ i.e. the area of a unit circle, it suffices to prove that ...
3
votes
2answers
57 views

Integral of constant divided by polynomial and another constant

$$\int_{-\infty}^{-1}\frac{4}{\sqrt{x^6+2}}\,dx$$ What are the steps to integrate?
0
votes
1answer
53 views

Integral of constant divided by polynomial and logarithm

$$\int_e^\infty\frac{2}{x-\ln(x)}\mathrm dx$$ I'm not sure how to integrate. What are your hints?
0
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0answers
10 views

Direct Comparison Test and error for improper integrals

problem: $\displaystyle\int_1^\infty {3+\cos(x) \, dx\over x^3}$ I used ${4\over x^3}$ as the comparison for the direct comparison test and obtained an answer of "two" giving me the conclusion that ...
0
votes
1answer
17 views

Proving convergence or divergence

I have to show the following diverges: $$\int_0^1 \frac{1}{x^{1/3} -x^{4/3}} dx$$ I am meant to do this without evaluating the integral. I know that I have to split it into: $$\int_0^{0.5} ...
8
votes
1answer
275 views

One difficult integral

Prove that $$\int_0^1x\log\left(1+x^2\right)\left[\log\left(\frac{1-x}{1+x}\right)\right]^3\operatorname{d}\!x$$ ...
1
vote
2answers
57 views

Definite Integral of $\int_{0}^{\infty}3^{-4(z^2)}dz$

Please help me insolving this $$\int_{0}^{\infty}3^{-4(z^2)}dz$$ I tried to do normal substitution but it didn'd work.... I wonder it is complex integration... I dont need the solution rather just ...
1
vote
1answer
43 views

Riemann sum from $0$ to $\infty$

I wonder which limit correctly defines the integral $I= \displaystyle\int_0^\infty u(x)dx $. Is it: $$ I = \lim_{h \rightarrow 0} \sum_{r=1}^\infty u(rh)h = \color{red}{\lim_{h \rightarrow 0}} ...
2
votes
3answers
58 views

Show that $\int_0^{\infty}\frac{f(ax)-f(bx)}{x}dx=[f(0)-L]\ln\frac{b}{a}$ [duplicate]

Let $f:[0,\infty)\to\mathbb{R}$ be continuous and $\lim_{x\to\infty}f(x)=L$. Show that $$\int_0^{\infty}\frac{f(ax)-f(bx)}{x}dx=[f(0)-L]\ln\frac{b}{a}$$ where $0<a<b$. I don't even know how to ...
9
votes
3answers
106 views

Evaluating this integral using the Gamma function

I was wondering if the following integral is able to be evaluated using the Gamma Function. $$\int_0^{\infty}t^{-\frac{1}{2}}\mathrm{exp}\left[-a\left(t+t^{-1}\right)\right]\,dt$$ I already have a ...
0
votes
3answers
79 views

Integration involving $\sin x/x$

Let $f$ be a differentiable function satisfying: $$\int_0^{f(x)}f^{-1}(t)dt-\int_0^x(\cos t-f(t))dt=0$$ and $f(\pi)=0$, Considering $g(x)=f(x)\forall x\in\mathbb R_0=\mathbb R+{0}$. If ...
1
vote
1answer
32 views

integration seems gaussian, but can't solve

i need some help in this integral $$\int_{-\infty}^{\infty}\exp(-bx^2) \frac{d^2}{dx^2} \left(\exp(-bx^2)\right) dx$$ I tried differentiating $\displaystyle e^{-bx^2}$ twice and it came up weird , ...
1
vote
4answers
34 views

Convergence of a function with $e$ in the denominator

$$\int^{\infty}_1\frac{dx}{x^3(e^{1/x}-1)}$$ I'm given the hint that the function $y = e^x$ has a tangent $y=x+1$ when $x=0\land y=1$. How do I prove its convergence and find a upper-limit for the ...
0
votes
0answers
20 views

bessels functions ,multiple integration

I'm not able to solve this bessel's function and not able to compute this step by step. i have tried in for computing this but even i'm not able to do this.
2
votes
0answers
50 views

Is there a closed-form expression for this trigonometric Cauchy Principal Value-type integral?

Consider the following definite integral, $I(n; \theta)$. $$ I(n; \theta) = \int_{0}^{\pi} \frac{\cos(n\phi)}{\cos\phi-\cos\theta} d\phi \quad \text{where } n \in N $$ When $0 < \theta < \pi ...
9
votes
1answer
244 views

Other challenging logarithmic integral $\int_0^1 \frac{\log^2(x)\log(1-x)\log(1+x)}{x}dx$

How can we prove that: $$\int_0^1\frac{\log^2(x)\log(1-x)\log(1+x)}{x}dx=\frac{\pi^2}{8}\zeta(3)-\frac{27}{16}\zeta(5) $$
1
vote
1answer
38 views

Evaluate integral using Fourier analysis

$\int_0^\infty \frac{\cos (x)}{1+4x^2}\, dx$ $\int_0^\infty \frac{1}{(1+x^2)^2}\, dx$ There is no hint for these two questions. I think for Q2, since it's a square, I can use Plancherel ...
0
votes
0answers
27 views

Approximation of an infinite series using an integral.

For electric potential I have the following infinite series: $V=k_e\frac{q}{r}+2k_eq\sum_{n=1}^{\infty}\frac{1}{\sqrt{r^2+n^2a^2}}$ Taking the derivative with respect to r, I have the following ...
4
votes
5answers
115 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
52 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
1answer
44 views

Generalized Riemann Integral: Improper Version

Reference For a bounded nonexample of integrability see: Riemann Integral: Bounded Nonexample For a convergence theorem on integral see: Riemann Integral: Uniform Convergence For a comparison of ...
1
vote
1answer
31 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
77 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
191 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
64 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
43 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
26 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
27 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
22 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
59 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 ...
6
votes
0answers
130 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
25 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
311 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
84 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
51 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
72 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
47 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
57 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
41 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
243 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
14 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
44 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 ...