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
44 views

Improper integral of $\frac{1}{x} \sin \frac{1}{x}$

Let $f(x)= \frac{1}{x} \sin \frac{1}{x}$ when $x \in (0, 1]$and $f(0)=0$. Prove that $\int_{0}^1 f(x)dx$ exists. Can someone give me a hint to solution?
3
votes
1answer
53 views

Prove that $\int_{2}^{\infty} \frac{dx}{x^{1-\epsilon}\cdot \ln(x)^{\beta}}$ diverges for every $\beta$.

let $\beta,\epsilon\in \mathbb R$, such that $\epsilon>0$. prove that for every $ \beta$: $$\int_{2}^{\infty} \frac{dx}{x^{1-\epsilon}\cdot \ln(x)^{\beta}}$$ Diverges. SOLUTION ATTEMPT: if ...
2
votes
1answer
42 views

Elliptical Integral that diverges at one point

I have to solve the following integral $$I=\int_{\lambda_1}^yd\lambda\frac{1}{1-\lambda}\sqrt{\frac{(\lambda-\lambda_1)(\lambda-\lambda_2)(\lambda-\lambda_4)}{\lambda-\lambda_3}}$$ where ...
2
votes
1answer
63 views

Closed form of $\int_{x = 0}^{C} \exp\left(-\frac{x}{A}-\frac{B}{x}\right)\,dx$

Is there a closed-form expression for the following definite integral? \begin{equation} \int_{x = 0}^{C} \exp\left(-\frac{x}{A}-\frac{B}{x}\right)\,dx, \end{equation} where $A$, $B$, and $C$ are ...
3
votes
2answers
75 views

Is there a simple proof for $\int_1^{\infty}\frac{2x^2\log^2 x}{(x^2-1)^2}dx=\frac{1}{4}(7\zeta(3)+\pi^2)$?

This morning I've computed easy computations with simple integral representations for Apéry constant and I find a (conjecture) formula using an online integrator (Wolfram Alpha), I woluld like if it ...
2
votes
0answers
47 views

Improper definite integral $\int_0^\infty e^{-x^2}(x + k)^\alpha dx$

I am unsure how to calculate the following definite integral: $$ \int_0^\infty e^{-x^2}(x + k)^{\alpha}dx,$$ where $k > 0$ and $\alpha$ is a real number. I tried integrating by parts and also a few ...
2
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2answers
39 views

Improper integral involving trigonometric function

I was wondering what happens when evaluating an improper integral involving a trigonometric function where the denominator is a rational function with a zero at $x=0$. The example I have in mind is ...
1
vote
2answers
63 views

Upper bound of $\int_{-\infty}^{\infty}\sin(x)dx$. [duplicate]

From another question, improprer integral $$\int_{-\infty}^{\infty}\sin(x)dx$$ is not $$\lim_{a \to \infty} \int_{-a}^a \sin x \, d x$$ but $$\lim_{a \to \infty}\lim_{b \to \infty}\int_{-a}^b \sin x ...
0
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0answers
19 views

Improper integrals $\int_{-\infty}^{+\infty} \cos t dt$ and $\int_{-\infty}^{+\infty} \sin t dt$ [duplicate]

What can we say about following improper integrals? $$\int_{-\infty}^{+\infty} \cos t dt, \ \ \ \ \ \ \int_{-\infty}^{+\infty} \sin t dt$$ My attempt: $$\int_{-\infty}^{+\infty} \cos t dt=2 ...
0
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3answers
88 views

How does the integral $\int_0^{\infty} \ln\left( 1+\frac{1}{x^2} \right) \,\text{d}x$ converge?

I tried using the fact that $\ln(f(x)) < f(x)$ but that doesn't seem to work. It's an improper integral. $$ \int_0^{\infty} \ln\left( 1+\frac{1}{x^2} \right) \,\text{d}x $$
2
votes
5answers
78 views

How to show $\int_{1}^{\infty} \frac{\sin^2(x)}{x^2}dx$ is finite?

At first, my approach was to directly take the improper integral of it. However, it seems not that easy. Then I tried to find another fraction to make a comparison. I got $\frac{\sin^2(x)}{x^2} ...
1
vote
1answer
54 views

Related integral problem to the Gaussian integral

So according to Proving $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx = \dfrac{\sqrt \pi}{2}$, $$\int_0^\infty e^{-x^2}dx=\frac{\sqrt{\pi}}{2}$$ I want to solve for this. $$\int_0^\infty e^{-x^2}\ln(x)dx$$ ...
2
votes
2answers
86 views

How do I evaluate $\int_{0}^{\infty} u^{z-1}(e^{iu}-1) \, du$?

I am trying to evaluate the following integral that shows up in this paper http://arxiv.org/pdf/1103.4306v1.pdf $I=\int_{0}^{\infty} u^{z-1}(e^{iu}-1)du= \Gamma(z)e^{\frac{iz\pi}{2}}$ for ...
2
votes
0answers
52 views

$1 = \int f(x) \ dx$, by definition, or by Lebesgue's theorem?

We have that (in the context of Lebesgue integration)$$\lim_{n \rightarrow \infty} \int_{-\infty}^n f(x) \ dx = 1$$ I wish to show that this implies $\int_{-\infty}^\infty f(x) \ dx = 1$. Is this true ...
0
votes
0answers
22 views

A multiple of the Gamma Function when integrated between $0$ and $\infty$

I was reading through this answer on stats.stackexchange, but didn't follow the mathematics behind one step. They have $$\int\limits_{\tau=0}^{\infty} e^{-\tau( ...
0
votes
0answers
25 views

Maximizing an Integral Quantity

Consider the function $f(x) = x^{-\ln x}$. Let $U(x)$ be a function such that $U'(x) > 0$ and $U(x) < x$. Suppose that $\int_0^\infty U(f(x)) = T$. What function $U$ maximizes the quantity ...
0
votes
1answer
28 views

Seeking possibility of more elementary means of evaluating an improper integral.

It can be shown that $\int_0^\infty -\log{(1-e^{-x})}=\zeta(2)$ by expanding out the integral as $\log(1-z)$, exchanging summation and integration, then summing up the integrals. I am wondering if ...
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1answer
28 views

Test for convergence of integral

Considering the behaviour of the integrant at both integration limits, study the convergence of the integral: $$\int_{0}^\infty x \sin\left(\frac{1}{x^\frac{3}{2}}\right). $$ I was trying to ...
0
votes
1answer
30 views

Same integral diverges for different limits?

I was investigating convergence of such an integral: $\int_{1}^\infty $$\frac{dx}{x(1+x)} $ I used comparison test: $\int_{1}^\infty $$\frac{dx}{x(1+x)} $ < $\int_{1}^\infty $$\frac{dx}{x^2} $ ...
2
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0answers
28 views

Integral of $\sin (e^{x^2})e^{-x^2+ix\lambda}$

Trying to solve this problem : Is $T$ invariant under Fourier transform ? Where : $T= \{f\in \mathcal{C}^{\infty} (\mathbb{R}), \forall n \in \mathbb{N}, |x|^nf(x) \to 0 \; \text{when}\; |x| \to ...
0
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1answer
32 views

Convergence and estimate of improper integral involving function $\frac{\sin\pi x}{\pi x}$.

Study the convergence of the improper integral $$\int_{-\infty}^\infty \frac{d^k}{dx^k}\left\{\frac{\sin(\pi(x-t))}{\pi(x-t)}\right\}\Biggl|_{x=n}\ \cdot \ ...
2
votes
2answers
95 views

How can we compute $\int^{\infty}_0\sin(x^2)dx$ using Fourier transform?

How can we compute $\int^{\infty}_0\sin(x^2)dx$ using Fourier transform? I had an idea in my mind. To use the $\text{sinc}$ function and take its inverse Fourier Transform or something like that. ...
0
votes
1answer
41 views

Determine if the improper integral converges

I'm having trouble proving that this improper integral converges, if it does. $$\int_3^{\infty} \frac{dx}{x+e^x}$$
0
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1answer
71 views

When is $\int_1^\infty \frac{x+\sqrt{x+\ln(x+2)}}{(x^a+\cos x)^{1/3}}\,\mathrm{d}x$ finite?

How do I solve the following problem? For which of the following values of $a$ is the integral $$ \int_1^\infty \frac{x+\sqrt{x+\ln(x+2)}}{(x^a+\cos x)^{1/3}}\,\mathrm dx $$ finite? $$ a ...
0
votes
5answers
99 views

Evaluating the integral $\int_{-\infty}^\infty x^2\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2} \ dx$

How does one evaluate $$\int_{-\infty}^\infty x^2\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2} \ dx ?$$ The result is $1$ and it corresponds to $E[X^2]$, where $X$ is a random variable with ...
1
vote
2answers
47 views

How do I change integration order?

I need to bound (or calculate in a closed form) this integral: $$\int_{R}^{\infty}dz\int_{a-bz}^{\infty}dy\cdot \frac{1}{\sqrt{2\pi}}e^{\frac{-y^2}{2}} e^{-z}$$ as a function of $R,a,b$. The result ...
0
votes
0answers
30 views

Integrating this (rather messy) integral and determining the limiting behaviour

From a preceding question I have worked on, an integral to evaluate: $$\frac{2i\sqrt{a}}{(a^2+1)}\cdot\int_{-\infty}^\infty k\cdot\exp\bigg[i\bigg(k-\frac{k^{2}ht}{2m}\bigg)\bigg]~dk$$ I need to ...
1
vote
1answer
29 views

Integrating an integrand with an absolute value on exponential

This is one heck of an embarrassment but it is amazing how these bits of subtlety gets lost in the back of the head after the first year of undergraduate studies-with every computation chucked into ...
3
votes
2answers
57 views

Is the improper integral $\int_0^{\pi/2} \sqrt{\cot x}\, dx$ convergent?

Is the improper integral $\int_0^{\pi/2} \sqrt{\cot x} \,dx$ convergent? I am unable to use any kind of comparison test or anything.
3
votes
1answer
87 views

Evaluating the improper integral $\int_0^{\infty}\frac{dx}{1+x^3}$

Evaluate $$\int_{0}^{\infty}\frac{dx}{1+x^3}.$$ I tried integration by partial fraction. My work is below: $$\int_{0}^{\infty}\frac{dx}{1+x^3}=\frac{1}{3} ...
3
votes
1answer
46 views

Convergence of an integral $\int_1^{+\infty} \frac{\ln^2(1+x)}{x^{2\alpha}}\mathrm dx$

$$\ln^2(1+x)\sim x^2-x^3,x\rightarrow \infty\Rightarrow \int_1^{+\infty} \frac{\ln^2(1+x)}{x^{2\alpha}}\mathrm dx=\int_1^{+\infty} \frac{x^2-x^3}{x^{2\alpha}}\mathrm dx=$$ $$\int_1^{+\infty} ...
2
votes
0answers
62 views

Any good approximation for this integral?

I am interested in the following integral $$ \mathcal{I}=\int_{-\infty}^\infty\mathop{dz}\left[\frac{1}{\sqrt{a+b}(z^2)^{n/4}}-\frac{1}{\sqrt{a+b\cos^2\theta}(R^2+z^2)^{n/4}}\right], $$ where $R\ll ...
3
votes
1answer
72 views

How to calculate Lebesgue integral in this type?

For Lebesgue integral in this type, like$$\int_\pi^\infty \left({1 \over {x \sin^{1/3}x} }\right)^2$$ can anyone give me some general idea? I don't know use which inequalities to start with the this ...
4
votes
1answer
83 views

Is this equality $\lim_{x \to \infty} \int_0^x \frac{t^2}{2(e^t-1)}\mathrm{d}t=\lim_{n \to \infty}\sum_{k=1}^n \frac{1}{k^3}$ true?

Using a little program in Python, it looks true for at least two hundred digits after the comma, but I have absolutely no idea, how to begin. Any hint sould be appreciate. $$\lim_{x \to \infty} ...
3
votes
3answers
106 views

Question on reasoning for $\int_1^\infty\frac{\sin(x)}{x}dx$ to converge

I often saw a 'proof' that $\int_1^\infty\frac{\sin(x)}{x}dx$ converges: By integration by parts we get $$\int_1^\infty\frac{\sin(x)}{x}dx = \cos(1)-\int_1^\infty{\frac{\cos(x)}{x^2}}dx$$ and thus ...
0
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1answer
32 views

Solving $\iint\frac{x^2}{(8x^2+6y^2)^{\frac 3 2}}$ on the domain $8x^2+6y^2\leq 1$

I need to solve $\displaystyle\iint\frac{x^2}{(8x^2+6y^2)^{\frac 3 2}}$ on the domain $8x^2+6y^2\leq 1$. I recognise this is an improper integral, so we need a monotonic series of domains ...
1
vote
2answers
30 views

Confusion in the usage/property of Laplace Transform.

While proving that $$\int^{\infty}_0 \frac{\sin x}xdx$$ I saw the Laplace Transform proof. It used that $$\cal L\left\{\frac{\sin t}{t}\right\}=\int^\infty_0 \cal L\left\{\sin(t)\right\}d\sigma$$ So ...
4
votes
1answer
50 views

Convergence of improper integral with $f(x)\to 1$ as $x\to +\infty$

Suppose $f\in \mathscr{R}$ on $[0,A]$ for all $A<\infty$, and $f(x)\to 1$ as $x\to +\infty$. Prove that $$\lim \limits_{t\to 0}t\int_{0}^{\infty}e^{-tx}f(x)dx=1 \quad (t>0).$$ Proof: Let's ...
3
votes
3answers
223 views

Solving $\int_{0}^{\infty} \frac{x^2}{x^8+5}dx$? [closed]

$$\int_{0}^{\infty} \frac{x^2}{x^8+5}\ \mathrm{d}x?$$ It's very hard for me to solve above integral. Please help me.
0
votes
1answer
96 views

Evaluate $\int_{0}^{1}{x\ln x dx}$.

Evaluate $\displaystyle\int_{0}^{1} x \ln x\, dx$. I used integration by parts with $u=\ln x$ and $dv=x\,dx$. Then I got this: $\displaystyle\frac{x^2 \ln x}{2}- \int_{0}^{1}\frac{x}{2}\,dx$. ...
1
vote
1answer
42 views

Test convergence, find $\alpha$ which makes integral converge

I'm testing the convergence of this improper integral $$\int_2^{\infty} x(\ln x)^{\alpha} dx$$ I used the limit comparison test with $\frac{1}{x}$ which is divergent, I found that this integral ...
12
votes
4answers
813 views

A mathematical fallacy concerning the integrability of a function and cancellation

I am reading the Florida Mu Alpha Theta Sponsors Guide. Page 43 is a list of clarifications and disputes commonly made, and their resolutions. One of their clarifications is this: A function ...
5
votes
5answers
122 views

Prove This Inequality ${\pi \over 2} \le \sum_{n=0}^{\infty} {1 \over {1+n^2}} \le {\pi \over 2} + 1$

$${\pi \over 2} \le \sum_{n=0}^{\infty} {1 \over {1+n^2}} \le {\pi \over 2} + 1$$ I see I should use Riemann sum, and that $$\int_0^{\infty} {dx \over {1+x^2}} \le \sum_{n=0}^{\infty} {1 \over ...
6
votes
0answers
57 views

Questions regarding $\int_0^\infty\frac{\sin^2x}{x^2}dx$ [duplicate]

question: $\int_0^\infty\frac{\sin^2x}{x^2}dx$ is equal to (A)$\int_0^\infty\frac{\sin x}{x}dx$ (B)$\int_0^\infty\frac{\cos x}{x}dx$ (C)$\int_0^\infty\frac{\cos^2x}{x^2}dx$ ...
0
votes
2answers
62 views

Is this integral less than infinity?

Assume the following integral: $$ \int\limits_{-\infty}^{\infty}\frac{f\left(x\right)} {BB\left(\lceil abs\left(x\right)\rceil\right)}\mathrm{d}x $$ Where $f\left(x\right)$ is any computable ...
5
votes
2answers
79 views

Improper integral of $\log x \operatorname{sech} x$

How to prove the following? $$ \int_0^\infty \log x \operatorname{sech}x\,dx = \frac{\pi}{2} \log\left( \frac{4\pi^3}{\Gamma(1/4)^4} \right) $$ I obtained the right side with CAS. It seems like this ...
0
votes
0answers
49 views

Improper integral of $\frac{\sin (\pi a t) \sin (\pi b t)}{t^2}$ depends only on the smaller of $a,b$ [duplicate]

$$\int_{-\infty }^{\infty } \frac{\sin (\pi a t) \sin (\pi b t)}{t^2} \, dt = \pi ^2 b$$ if $a$ and $b$ positive and $a>b$ (from Mathematica). How is this result only based on the smaller of ...
2
votes
1answer
24 views

Proof for the behavior of both types of improper integrals for different powers of x

I was trying to prove for what values of p eq.1 converges or diverges, they didn't give the proof for eq.1 but for eq.2 a proof was given and when I was done with the proof for eq.1 I noticed that for ...
0
votes
3answers
104 views

Properties of improper integral (showing that: $\int \limits_{0}^{\infty}f(x)dx=\int \limits_{0}^{1}f(x)dx+\int \limits_{1}^{\infty}f(x)dx.$)

Let $f(x)$ is integrable on every segment $[r,\infty)$ where $r>0$. Let $\int \limits_{0}^{1}f(x)dx$ and $\int \limits_{1}^{\infty}f(x)dx$ converges. Why in this case we can conclude that $$\int ...
0
votes
0answers
57 views

Integrate $\int^1_{0} \frac{\ln (x+1)}{x^2+1}dx$ [duplicate]

$$\int^1_{0}\frac{\ln (x+1)}{x^2+1}dx$$ I'm having trouble solving this one. I tried trigonometric subst. but that doesn't get me far:$$\int^1_{0}\frac{\ ln (tan\theta+1)}{\sec^2\theta}\sec^2\theta ...