Questions tagged [improper-integrals]

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 $\infty$ or $-\infty$, or as both endpoints approach limits.

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For nonnegative function, does $\int_0^\infty f(x)dx$ converge or diverge to $\infty$?

Let $f$ be a nonnegative function defined on $[a,\infty)$. Consider $\displaystyle\int_a^\infty f(x) dx=\lim_{M\to \infty}\ \int_a^M f(x) dx$. Let $\mathcal F:=\left\{\displaystyle\int_a^M f(x) dx \...
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Are the functions $1 - \frac{2}{\pi}\tan^{-1}(x^2)$ and $e^{-x^2}$ related?

Here is a graph of these functions: https://www.desmos.com/calculator/8opjgf6vxu They are almost identical when compared graphically. I'm wondering if there's any modifications (horizontal/vertical ...
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9 votes
2 answers
141 views

Ultimate Sophomore's dream $\int_0^1\cdot\cdot\cdot\int_0^1 (t_1t_2\cdot\cdot\cdot t_n)^{t_1t_2\cdot\cdot\cdot t_n} dt_1 dt_2\cdot\cdot\cdot dt_n$

$$I_n=\int_0^1\int_0^1\cdot\cdot\cdot\int_0^1 (t_1t_2\cdot\cdot\cdot t_n)^{t_1t_2\cdot\cdot\cdot t_n} dt_1 dt_2\cdot\cdot\cdot dt_n$$ Prove: $$I_1=I_2<I_3<\cdot\cdot\cdot <1=\lim_{n\...
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  • 335
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How can I proceed with the convergence of an improper integral

I am having some trouble with this integral right here $$\int_0^1\sqrt{x}\sin\left(\sqrt{x}\right)e^{3\sqrt{x}}\left(\frac{x}{1-x^2}\right)^{\alpha-1}dx$$ I know that the problems lies when $x=1$, ...
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1 vote
0 answers
19 views

Bound the integral $\int_{-\infty}^{\infty} \frac{f(a+it)}{(1-ist)^n} dt$

For $$ f(x) = \frac{x}{(x+1)(x+2) - e^{-3x}} $$ I try to prove that there exist $M > 0$ such that for all non-negative integers $n$ and all $x > 0$ we have $$ |f^{(n)}(x)| \le M n! \frac{1}{x^{...
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-3 votes
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How to find or approximate this integral? [closed]

How can we calculate or approximate this integral? $$\int_{\alpha_j} \exp\left(\hspace{0.6mm} - \frac{(\alpha_1...\alpha_J) (\Sigma^{(z)})^{-1} (\alpha_1...\alpha_J)^T }{2} + \sum_{j=1}^J ( - P\exp(\...
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1 vote
0 answers
32 views

An improper integral involving a modified Bessel function, an exponential and hyperbolic functions.

Let $ x>0 $ and $a \in {\mathbb R}$, $b \in {\mathbb R} $. We consider a following improper integral. We have: \begin{equation} {\mathfrak J}_{a,b}(x):= \int\limits_0^\infty e^{-2 x \sinh(\omega)} \...
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  • 9,113
0 votes
2 answers
69 views

Generalizing $\int_{0}^{\infty }\frac{f(t)}{t}dt=\int_{0}^{\infty }\mathcal{L}\left \{ f(t) \right \}ds$.

Here, I saw the following formula: $$\int_{0}^{\infty }\frac{f(t)}{t}dt=\int_{0}^{\infty }\mathcal{L}\left \{ f(t) \right \}ds$$ Say we have the integrand only $f(t)$, not $\frac{f(t)}{t}$, then I ...
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4 votes
1 answer
98 views

Evaluating integral $\int_{-\infty}^\infty \frac{x}{\sqrt{x^2 + c^2}} e^{-(x-m)^2} dx$ and one more similar

I'm trying to evaluate two integrals: $$f(m,c) = \int_{-\infty}^\infty \frac{x}{\sqrt{x^2 + c^2}} e^{-(x-m)^2} dx$$ and $$g(m,c) = \int_{-\infty}^\infty \frac{1}{\sqrt{x^2 + c^2}} e^{-(x-m)^2} dx$$ ...
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  • 165
2 votes
1 answer
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To show $\int_{0}^{\infty} \frac{x}{1+x^6\sin^2x}\,dx$ is convergent

Show that $\displaystyle \int\limits_{0}^{\infty} \frac{x}{1+x^6\sin^2x}\,dx$ is convergent Since we need to check if the given improper integral is convergent, I considered Abels/Dirichlet test for ...
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0 votes
0 answers
31 views

Fractional powers integral formula [duplicate]

I found this formula for the case of operators in H.O.Fattorini's book Second Order Linear Differential Equations in Banach Spaces. page 51.$$(-a)^\alpha=\frac{\sin(\alpha \pi)}{\pi} \int_0^\infty \...
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1 vote
1 answer
48 views

Integral $\int_{-\infty}^0 e^{-t^2/2}dt$

can someone help with this integral $\int_{-\infty}^0 e^{-t^2/2}dt$? I know that $\int_{-\infty}^{\infty} e^{-t^2/2}dt = \sqrt{2 \pi}$ and $\int_{-\infty}^{\infty} e^{-t^2}dt = \sqrt{\pi}$. I tried to ...
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  • 11
1 vote
1 answer
38 views

Numerical integration of a function with a singularity

I'm trying to compute numerically a function like the following: $$ F(t)=\int_{0}^{t}{\frac{f(\tau)}{\sqrt{t-\tau}}d\tau} $$ I tried to adopt the composite Simpson's rule, but the problem is that when ...
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0 votes
1 answer
23 views

improper integral - prove divergence or convergence

prove if: $\int _{R^2}e^{-(x+y)^2} dx dy$ diverge or converge I have proven that $\int _{R^2}e^{-x^2-y^2} dx dy$ converge (and equal to $\pi$) using polar coordinates, but here the exponent of $e$ ...
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  • 55
1 vote
0 answers
53 views

Meaning of the integral $I(t)=\lim_{\epsilon\rightarrow 0^+} \int_0^t\frac{\phi(y)}{(\epsilon +iy)^n}dy$

I am trying to give a meaning to this integral $$I(t)=\lim_{\epsilon\rightarrow 0^+} \int_0^t\frac{\phi(y)}{(\epsilon +iy)^n}\,dy$$ where, $n > 1$, $\phi(y)$ is a complex function, infinitely ...
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0 votes
1 answer
58 views

Complicated improper integral

Compute $$\int_0^\infty \frac{x}{(1+x^2)x^\alpha}dx, \space \alpha \in (0,1).$$ So I thought of writing it as $$\int_0^\infty \frac{x^{1-\alpha}}{1+x^2}dx$$ But since $\alpha \in (0,1)$ a simple $u$-...
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6 votes
0 answers
221 views

Is it true that $\int_1^{\infty}\frac{{dy}\,/\,{y^2}}{1+(y-1)^{\alpha}}=\frac12$ for any real number $\alpha$? [closed]

This is an amusing integral evaluation. QUESTION. Is this true? For any real number $\alpha$, we have $$\int_1^{\infty}\frac{{dy}\,/\,{y^2}}{1+(y-1)^{\alpha}}=\frac12.$$
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0 votes
1 answer
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Looking for a function with specific properties

I am looking for a differentiable function $f$ that satisfies all of the following: $$(\text{I}) \lim_{x\rightarrow 0^{+}}f(x) \rightarrow \infty$$ $$(\text{II}) \lim_{x\rightarrow \infty}f(x) \...
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0 votes
0 answers
25 views

For which values of $\alpha$ does the integral $\int_{||x||\geq1} \frac{dx}{||x||^\alpha}$ converge?

Let $n\in \mathbb{N}$. I want to determine the values of $\alpha>0$ such that: $$I_n(\alpha):=\int_{\mathbb{R^n}\setminus B_1(0)}\frac{1}{||x||^\alpha}dx<+\infty$$ Clearly these are dependent on ...
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3 votes
2 answers
116 views

Problem with the integral $\int_0^{+\infty} \frac{t^{m-1}}{1+t^{2n}}{\rm d}t$

I'd like to prove, using a partial fraction decomposition (I don't want to use residue calculus), that $$\int_0^{+\infty} \frac{t^{m-1}}{1+t^{2n}}{\rm d}t=\frac{\pi}{2n\sin\frac{m\pi}{2n}}$$ where $1\...
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0 votes
0 answers
63 views

Integral of product of modified Bessel functions: $ \int_0^{\infty} r e^{- a r^2} I_k (b r^2) I_{2k-n} (c r) d r $

The following integral appeared in my research recently. $$ \int_0^{\infty} r e^{- a r^2} I_k (b r^2) I_{2k-n} (c r) d r , \tag{*} $$ where $c \geq 0$, $a > b \geq 0$, and $k , n \in \mathbb{Z}$. ...
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0 votes
0 answers
34 views

If $f$ is strictly increasing and integrable on $[0,+\infty)$ then $f(x)<0$ for any $x \in [0,+\infty)$

Prove that if $f$ is strictly increasing and integrable on $[0,+\infty)$, then $f(x)<0$ for any $x \in [0,+\infty)$. I tried this: assume by contradiction that there exists at least one $x_0 \in [...
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  • 391
4 votes
2 answers
152 views

Definite integral $ \int _{0}^{\infty } x\cdotp \tanh( 2x) \cdotp \ln(\coth x)\mathrm{d} x$

I want to show that $\displaystyle \int\limits _{0}^{\infty } x\cdotp \tanh( 2x) \cdotp \ln(\coth x)\mathrm{d} x=\frac{\pi ^{2} \cdotp \ln( 2)}{2^{4}}\tag*{}$ I tried integration by parts, Feynman ...
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1 vote
2 answers
157 views

Let $\lim_{n\to \infty} \sum_{r=1}^n \frac{1}{r^2}=\frac{\pi^2}{6},$ then which of the following is/are true?

For any natural number $n,\;$ Let $$\lim_{n\to \infty} \sum_{r=1}^n \dfrac{1}{r^2}=\dfrac{\pi^2}{6},$$ Then which of the following is/are true? (A) $$\int_{0}^{1} \dfrac{\ln( 1+x)}{x}dx=\dfrac{\pi^2}{...
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  • 2,050
0 votes
0 answers
30 views

Limit with an improper integral

Is it true that $$\displaystyle\lim_{x\to 0}x\int_{0}^\infty k^5 dk=0$$ Seems like this has an indeterminate form, but I am being told it equals zero.
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0 votes
1 answer
60 views

Convergence of the integral of $x\sin(x^p)$ from $0$ to $+\infty$

Let $p\in \mathbb{R}, p>0$ for which values of $p$ does the following integral converge? $$\int_0^{\infty}x\sin(x^p)dx $$ I'm not sure how I can go about this question, I know that simply trying ...
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  • 576
-2 votes
0 answers
41 views

Why Does $\int_0^\infty{x^{-ln x}dx} = \sqrt[4]{e}\sqrt{\pi}$? [duplicate]

I was curious what this integral converged to, so I checked on Wolframe Alpha and it told me $\sqrt[4]{e}\sqrt{\pi}$. The problem is, I don't know why that's the answer and haven't had luck finding ...
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0 votes
1 answer
57 views

Upper bound improper integral in terms of only the integrand

Suppose $f:[0,\infty)\to[0,1]$ is a nonincreasing function with $f(0)=1$, $\lim_{x\to\infty}f(x) = 0$, and also suppose that $f$ is integrable, i.e. $\int_0^\infty f(x)dx < \infty$. I am interested ...
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  • 9
0 votes
1 answer
21 views

Using a definition of an improper double integral over the xy-plane as a square whose sides increase to infinity

Textbook problem (quoted): An equivalent definition of the improper integral in part (a) is $$ \iint_{\mathbb{R}^{2}} e^{-\left(x^{2}+y^{2}\right)} d A=\lim _{a \rightarrow \infty} \iint_{S_{a}} e^{-\...
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3 votes
1 answer
83 views

Calculate double integral $\iint_D\frac{\sqrt{x^2+y^2 - a^2}x}{(x^2+y^2)^2}dxdy$ over unbounded region $D$

Calculate $$I = \iint\limits_D\frac{\sqrt{x^2+y^2 - a^2}x}{(x^2+y^2)^2}\,dx\,dy$$ where the region of integration is: $$D = \{(x,y) \in \mathbb{R}^2 \mid x+y - a\sqrt{2} \geq 0, -x+y +a\sqrt{2} \geq 0,...
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  • 79
0 votes
1 answer
88 views

A Solution for $\int_{0}^{\frac{1}{a}} x^2e^{-ax} \: dx$ without Integration by Parts

I came across this integral while doing some physics. I'm familiar with the trick where you can solve the integrals $$\int_{0}^{\infty}x^2e^{-ax} \: dx \: \: \: \text{or} \:\int_{0}^{\infty}x^2e^{-ax^...
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  • 3
2 votes
1 answer
86 views

Let $a>0$. Prove the improper integral $\int_0^{\infty} \cos\left\{{a\over2}\left(x+{1\over x}\right)\right\}x^{k-2}\ dx$ converges for $k>2$

There is a hint which says $\left|\int\limits_0^{\infty} \cos\left\{{a\over2}\left(x+{1\over x}\right)\right\}x^{k-2}\ dx\right|\le C a^{k-2}$ where C is some constant. I somehow feel that I need to ...
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0 votes
1 answer
39 views

How to show that the integral $\int_{0}^ {\infty} \frac{x^n}{(1+x)^m}dx$ converge when $m > n+1$ when $m,n$ are both positive integers?

How to show that the integral $\int_{0}^ {\infty} \frac{x^n}{(1+x)^m}dx$ converge when $m > n+1$ when $m,n$ are both positive integers? I have tested this for specific numbers and it looks like we ...
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  • 175
1 vote
2 answers
97 views

The closed form of $\int_{0}^{\infty} \frac{\cos x}{\left(x^{2}+1\right)^{n}} d x$

We are going to deal with the integral $$ I_{n}=\int_{0}^{\infty} \frac{\cos x}{\left(x^{2}+1\right)^{n}} d x $$ by differentiation on its related integral $$ J(a)=\int_{0}^{\infty} \frac{\cos x}{x^{2}...
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  • 5,480
0 votes
2 answers
87 views

How to find the exact value of $\int_{-\infty}^{\infty} \frac{\cos \pi x}{x^{2}-2 x+2} d x$?

I am going to evaluate the exact value of the integral $$ \int_{-\infty}^{\infty} \frac{\cos (\pi x)}{x^{2}-2 x+2} d x $$ using contour integration along anti-clockwise direction of the path $$\gamma=\...
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  • 5,480
0 votes
0 answers
36 views

Integral involving multiple Bessel functions.

I am trying to see if there are any known solutions to integrals of the following form, any help would be appreciated: $$\int_0^\infty xJ_{\nu+1}(ax)J_{\nu}(bx)\sin(cx)dx.$$ I've dug through a few ...
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0 votes
1 answer
62 views

How to find an equivalent of $\int_{1}^{+\infty}\exp{(-x^n)}dx$?

It's an exercise, I need to find an equivalent of : $$\int_{1}^{+\infty}\exp{(-x^n)}dx$$ I tried this : Let $x=u^{1/n}.$ Then $dx=\frac{1}{n}u^{1/n-1}\,du,$ so : \begin{align} \int_1^{+\infty} e^{-x^n}...
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0 votes
1 answer
59 views

Improper integral $\int_0^{2020} \frac{\sqrt{x}\arctan{x^\frac{3}{2}}}{\ln(1+x^2)\sin(\sqrt{x})}dx$

I have to investigate convergence of improper integral $$\int_0^{2020} \frac{\sqrt[3]{x}\arctan{(x^\frac{3}{2}})}{\ln(1+x^2)\sin(\sqrt{x})}dx$$ Singularities are $k^2\pi^2$ for $k\in\mathbb{Z}$. ...
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  • 6,192
0 votes
0 answers
38 views

Convergence of non-integer order Bessel addition formula and Schlafli's integral

I'm having some issues of seeing the derivation of the non-integer order Bessel function product relations. They're given as follows, $$ J_v(\alpha\pm \beta)= \sum_{n=-\infty}^{\infty}J_{v \mp n}(\...
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2 votes
1 answer
55 views

Improper integral $\int_0^1\frac{\log{(1-x^2)}}{(a-x)^2}dx$

It seems that integral converges for $a>1$, but I have problem with singular point $1$. What happens in case $0<a\leq 1$? Also, is it possible to calculate this integral (I tried with partial ...
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  • 6,192
1 vote
0 answers
33 views

Improper integral with two parameters

I have to investigate convergence of integral $\int_1^\infty\frac{\ln^q{x}}{(x^2-p)\sqrt{x^2+3}}dx$. For $p<1$ integral converges. For $p=1$, after substitution $t=x-1$, integral can be written as ...
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  • 6,192
0 votes
0 answers
76 views

Proving $\int_{-\infty}^{\infty} f = \lim_{a\to\infty}\int_{-a}^a f$

I have to show that Let $f$ be locally integrable and nonnegative on $(-\infty, \infty)$. Then $$\int_{-\infty}^{\infty} f = \lim_{a\to\infty}\int_{-a}^a f$$ The integral here is improper Riemman ...
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  • 109
0 votes
1 answer
38 views

Improper integral $\int_e^\infty \frac{dx}{\sqrt{(x^2-e^p)(\ln^3{x}-1)}}$

Convergence of improper integral $$\int_e^\infty \frac{dx}{\sqrt{(x^2-e^p)(\ln^3{x}-1)}}$$ has to be investigated. After substitution $x=et$, I got integral $$\int_1^\infty \frac{dt}{\sqrt{(t^2-e^{p-2}...
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  • 6,192
0 votes
1 answer
29 views

Convergence of $\int _4^\infty \frac{\ln ^{a+1}(x-3)}{\sqrt{e^{x-4}-1}}\,dx$

I have to investigate convergence of improper integral $\int_4^\infty \frac{\ln ^{a+1}(x-3)}{\sqrt{e^{x-4}-1}}\,dx$. After substitution $t=x-4$, I got that integral behaves as $\int_0^\infty \frac{\...
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  • 6,192
1 vote
2 answers
175 views

Show the inequality $\frac{\sqrt{\pi}}{2}<\left(\pi-e\right)!$

Working a bit on About the inequality conjectured as $x!>\left(\arctan\left(\cosh\left(x\right)\right)\right)^{a}$ for $x>0$ and fixed $a$ I got the inequality: $$\frac{\sqrt{\pi}}{2}<\left(\...
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  • 3,284
0 votes
1 answer
67 views

$\int_{S_r} \frac{e^{iz}-1}{z^2} dz$ as $r\rightarrow 0$

In my attempt to calculate $\int_0^\infty \frac{\cos(x)-1}{x^2}dx$ I arrived at the following expression: $$\int_r^\infty \frac{\cos(x)-1}{x^2}dx=\int_{S_r}\frac{e^{iz}-1}{z^2}dz$$ whhere $S_r=re^{it}$...
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7 votes
1 answer
73 views

Calculating contour integral with keyhole contour [duplicate]

Evaluate the integral $\int_0^{\infty}\frac{\log(x)}{(x+a)(x+b)}dx$ with $a,b>0, a\neq b$ by using the keyhole contour. My attempt- I drew the keyhole contour (obviously it's supposed to say $-a,-...
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  • 2,621
0 votes
0 answers
34 views

Show $\int_0^\infty f(x) \cos(ax) dx \ge 0$ if $f$ is convex and $a \in \mathbb R$.

Let $f$ be a convex function and $a \in \mathbb R$. Is it true that $$\int_0^\infty f(x) \cos(ax) dx \ge 0$$ holds? Of course, this would be true if we replaced $\cos$ by $|\cos|$, but I guess that ...
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  • 318
0 votes
2 answers
80 views

Improper integral $\int_0^{+\infty} \frac{\arctan x}{\sqrt{x}} \sin(2x) dx$

Check absolute and conditional convergence of following improper integral $$\int_0^{+\infty} \frac{\arctan x}{\sqrt{x}} \sin(2x) dx$$ Can someone please check if my solution is alright. Here is what I ...
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  • 353
1 vote
0 answers
37 views

solution of infinite integral by Hankel transform or by other means

I am trying to find the solution of the integral: $$\int_a^{\infty}\left(\frac{d}{dt}Ai(t/3^{1/3})\right) J_0(n\sqrt{t^2-a^2})dt,$$ where $ Ai(t)$ is the Airy function and $J_0(t)$ is the zeroth order ...
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