Questions about the evaluation of specific definite integrals.

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Is the integral $I_t(x)=\int_{-\infty}^\infty e^{-\frac{\cosh^2(u+t)}{2x}}\,e^{-\frac{u^2}{2t}}\,\sin\left(\frac{\pi\,u}{2t }\right)\,du$ positive?

We have the following integral $$I_t(x)=\int_{-\infty}^{\infty}e^{-\frac{\cosh^2(u+t)}{2x}}\,e^{-\frac{u^2}{2t}}\,\sin\left(\frac{\pi\,u}{2t }\right)\,du\,\,\,\text{ with }\,\,\,x\,,\,t>0$$ I ...
2
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2answers
29 views

Evaluating integrals in R^m

Let $|\cdot|_m$ denote the Euclidean norm in $\mathbb{R}^m$. Then I wish to prove that $\displaystyle\int\limits_{\mathbb{R}^m}|x|_me^{-|x|_m}dx<\infty$ It's kinda embarrassing to say this, but ...
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1answer
24 views

evaluation of some integral

I'm looking for a direct proof of the following identity: $$\frac{1}{\pi}\int_1^x\frac{dt}{t\sqrt{t-1}}\arcsin\left(\sqrt{\frac{x-t}{y-t}}\right)=\arcsin\sqrt{\frac{x}{y}}-\arcsin\sqrt{\frac{1}{y}},$$ ...
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1answer
33 views

Closed form solution of $\int \exp(-a (b-x)^{3/2}-cx)\text dx$

Does following integral have a closed form solution (a, b, and c are constants) $$ \int \exp(-a (b-x)^{3/2}-cx)\text dx $$ If not possible, what about a function with close behavior. $$ \int \exp(-a ...
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2answers
32 views

Definite integral with trigonometric function

I have some problem when trying to solve $$ \int_0^{\pi} \frac{1}{1 + 3\sin^2x}dx $$ I know, that it is the same (because it is symetric) a $$ 2 \int_0^{\frac{\pi}{2}} \frac{1}{1 + 3\sin^2x}dx $$ but ...
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1answer
65 views

Can this definite integral be solved?If so, how?

$$\int_{\pi\over4}^{\pi\over2}\sqrt{-\cos(2x)}\,dx$$ I tried everything, from substitution over tangents to duplication trig functions, can anyone give it a try that believes they can do it?
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2answers
92 views

Physically impossible to find the constant

How can we show $$ g(a) = \int _a^{a+1} \left\{x\right\} \cdot \left(1 - \left\{x\right\} \right)\:dx = \mbox{const} $$ where $\{x\}$ is the fractional part of $x$.
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2answers
70 views

Impossible to prove that is unbounded!

How we can demonstrate that $$\int _1^e\:\left(1+\log\left(x\right)\right)^ndx$$ is unbounded as $n\to \infty$, without using Bernoulli's Inequality?
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1answer
45 views

Proving that $\int_0^\infty\frac{J_{2a}(2x)~J_{2b}(2x)}{x^{2n+1}}~dx~=~\frac12\cdot\frac{(a+b-n-1)!~(2n)!}{(n+a+b)!~(n+a-b)!~(n-a+b)!}$

How could we prove that $$\int_0^\infty\frac{J_{2a}(2x)~J_{2b}(2x)}{x^{2n+1}}~dx~=~\frac12\cdot\frac{(a+b-n-1)!~(2n)!}{(n+a+b)!~(n+a-b)!~(n-a+b)!}$$ for $a+b>n>-\dfrac12$ ? Inspired by ...
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1answer
36 views

Direct comparison test for ( Improper ) Integrals [on hold]

How we can prove with direct comparison test for ( Improper ) Integrals that is bounded: $\int _1^n\:e^{-x^3}dx$ ?
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1answer
32 views

Very difficult to prove a convergent with Weierstrass

How we can prove that is monotone and bounded: $I_n=\int _1^n\:e^{-x^3}dx\:$ , Have any ideea how we can solve? and explain all to understand, I am a student... P.S: for all guys on this site, you ...
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2answers
28 views

$F(t)=(\beta/\alpha)\int_{0}^{t}\frac{(t/\alpha)^{\beta-1}}{[1+(t/\alpha)^{\beta}]^2}dt$

Compute $$F(t)=(\beta/\alpha)\int_{0}^{t}\frac{(t/\alpha)^{\beta-1}}{[1+(t/\alpha)^{\beta}]^2}dt$$ My Attempt : Let $u=(t/\alpha)\Rightarrow \alpha du=dt$ so, ...
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0answers
12 views

Convolution Integral of superposition of exponential and chirp signal [on hold]

Someone can help me with this convolution integral calculation? Any assistance would be greatly appreciated, thank you.
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0answers
28 views

Change of variables in entwined integral

The specific problem I am trying to solve is $$\int_{-1}^0 dx_1 \int_{-1}^{x_1}dx_2 \cdots \int_{-1}^{x_n}dx_{n+1} f_1(\tilde x_1)\cdots f_{n+1}(\tilde x_{n+1})$$ and doing the change of variables ...
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3answers
96 views

Computing $\lim_{n \to \infty} \sqrt[n]{ \int_{0}^{1} (1+x^n)^n dx}$

I tried to compute the limit by using Newton's binomial formula, variable change, studying the function $1+x^n$ to establish an inequality, but failed to solve it. $$\lim_{n \to \infty} \sqrt[n]{ ...
2
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2answers
57 views

How to prove $ I_t=\int_{0}^{\infty}g(u)\,\cos\bigl(\frac{\pi\,u}{2t }\bigr)\,du$ is positive?

We have the following integral $$I_t=\int_{0}^{\infty}g(u)\,\cos\left(\frac{\pi\,u}{2t }\right)\,du\,\,\,\text{with}\,\,\,\,t>0$$ where $g(u)$ is continuous on $(0,\infty)$, $g(0)=1$ and $g(u)$ is ...
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3answers
82 views

Evaluate $\int_0^2 x^2 e^{-x^3} \, dx$ using substitution

The integral in question is: $$\int_0^2 x^2 e^{-x^3} \, dx.$$ I'm not sure if I should put $u=x^2$ and $du=2xdx$ which would lead to $(1/2)du=xdx$ or put $u=e^-x^3$ which I'm not sure would change ...
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1answer
38 views

differentiable on open interval zero elsewhere

When thinking about another question I came across this and wonder if it's true: If $F$ is differentiable on $(a, b)$ and continuous at $[a, b]$, then given $g (x) = F^{\prime} (x)$ on $(a, b)$ and ...
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0answers
40 views

Evaluate:$\int_0^{\pi/2}\frac{x \sin x \cos x}{(a^2 \cos^2 x+b^2 \sin^2 x)^2}dx$ [duplicate]

How to evaluate the integral of $$\int_0^{\pi/2}\frac{x\sin x\cos x}{(a^2 \cos^2 x+b^2 \sin^2 x)^2}dx$$
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1answer
35 views

Definite integral of $y=x$ and $y=\frac{1}{\sqrt{x}}$

So I got the following problem : Calculate the area of the domain between the curves $y=x, \; y=\frac{1}{\sqrt{x}}$ and the horizontal lines $y=1$ and $y=2$. I think that the answer is $\frac{1}{2}$. ...
2
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1answer
52 views

Intuition for relationship between $xe^{-x^2}$ and $\frac{1}{2}e^u$

I understand how to do the computation here: $$\int_{0}^{\infty}xe^{-x^2}dx = \frac{1}{2}\int_{-\infty}^{0}e^{u}du = \frac{1}{2}$$ But I don't have any intuition for why the right-hand area under ...
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2answers
70 views

How to do this definite integration?

I was asked $$\int_a^b\! \frac{dx}{\sqrt{(x-a)(b-x)}}$$ I was surprised when I checked my answer. I can't upload my solution as the picture size us more than 2mb. But I am getting ...
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1answer
13 views

Area of one turning of Archimedian spiral

So, if the Archimedian spiral is given with formula $r=2\theta$, what does that formula represent and what is the area of one turning of the spiral? The teacher solved it like: ...
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2answers
26 views

What's the area of one arch of a cycloid?

So, the cycloid is given with parametric equations: $$x=r(t-\sin{t})$$ $$y=r(1-\cos{t})$$ The teacher solved it like this: $$P=\int_a^by(x)dx$$ $x=x(t)$; $\alpha<t<\beta$ ...
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2answers
1k views

Integral involving an error function

For $\sigma>0$, how can we prove that $$\frac{1}{2}\int_{-1}^1 \text{erf}\left(\frac{\sigma}{\sqrt{2}}+\text{erf}^{-1}(x)\right) \, \mathrm{d}x= \text{erf}\left(\frac{\sigma}{2}\right)$$ where erf ...
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1answer
45 views

Changing limits of an integral

The integral: $$ \int_{-\infty}^0\frac{e^{i\alpha x}\,dx}{1+x+x^2}. $$ If I want to change the limits of this integral so that the integral is taken from $0$ to infinity instead of minus infinity to ...
3
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1answer
62 views

Evaluate a limit using integral

$$\lim _{n\to \infty \:}\left(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2n-1}-\frac{1}{2n}\right)$$ How I evaluate this limit?
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2answers
38 views

Prove the following equation(much needed help):

Let there be a given function $f \in C([0,1])$, $f(x)>0$; $x\in [0,1]$. Prove $$\lim_{n\to\infty} \sqrt[n]{f\left({1\over n}\right)f\left({2\over n}\right)\cdots f\left({n\over ...
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1answer
63 views

Does $\int_{0}^{1} |f(t)|dt=0$ imply that $f(t)=0\ \text{for all } t \in [0,1]$? [on hold]

$$\int_{0}^{1} |f(t)|dt=0$$ Does this equation imply that $f(t)=0$ for every $t \in [0,1]$? I need a proof of whether the answer is yes or no. I couldn't prove this; I started thinking, if it were ...
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2answers
66 views

Integral of series expansion…

Why is $$\int _0^1\:\left(1-x+x^2-x^3+x^4-....-x^{2n-1}\right)dx = \int _0^1\:\left(\frac{1-x^{2n}}{1+x}\right)dx,$$ why do we have $1+x$ in the denominator, and why does $1-x^{2n}$ appear in the ...
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3answers
27 views

To calculate the expectation of a variable with a given pdf

Let the pdf of a random variable X be given by $f(x)=ae^{-x^2-bx}, -\infty<x<\infty$. If $E(X)=-\frac{1}{2}$, then (A)$a=\frac{1}{\sqrt{\pi}}e^{-1/4},b=1$ ...
2
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1answer
61 views

Find the integral $\int_{-2}^{1} |x| d |x|$

To handle the integral $\int_{-2}^{1} |x| d |x|$ I do not see a way to begin with. If instead I am to compute $\int_{-2}^{1} |x| dx$ then things get easier, for simply computing $\int_{0}^{1} x dx + ...
6
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2answers
58 views

Help to resolve a Double Integral

I'm doing a workout guide about double integrals and I came across an exercise that I could not resolve for a while. $$\int_0^2\int_1^2 \frac{x}{\sqrt{1+x^2+y^2}} \,\mathrm dx\,\mathrm dy$$ I guess ...
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1answer
68 views

A Logarithm Integral II [on hold]

Does the integral \begin{align} \int_{0}^{1} (1-t)^{2} \, \ln^{k}(1-t) \, \ln^{m}(t) \, dt \end{align} have a compact form for $m = 1$, and $m=2$ ?
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4answers
66 views

Evaluate $\int_{0}^{1}(1-x)^ndx$ by expanding the bracket.

I'd like to get a hint on this exercise. I believe I'm somewhat close to the answer. I used the binomial theorem to get: $\displaystyle\int_{0}^{1}(1-x)^ndx = \int_{0}^{1}\sum_{k=0}^{n}{n\choose ...
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1answer
37 views

Prove $\int^a_0 x(a^2-x^2)^{\nu-1} I_0(b x)dx= 2^{\nu-1}a^\nu b^{-\nu}\Gamma(\nu)I_\nu(a b)$

How can I prove the following equality? $$ \int^a_0 x\left(a^2-x^2\right)^{\nu-1} I_0\left(b x\right)dx= 2^{\nu-1}a^\nu b^{-\nu}\Gamma\left(\nu\right)I_\nu\left(a b\right), $$ under the ...
2
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1answer
73 views

Closed form for an almost-elliptic integral

Does $$\int_0^{2 \pi} \log\left(\frac{1}{2}[1+\sqrt{1-(a \sin\phi)^2}]\right) d\phi $$ have a closed form ? An approximation for small $a$ is $2E-\pi$, but it is the exact form that is needed ...
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1answer
23 views

Rearranging double integral and bounds

I am trying to figure out why we can rewrite $\int_0^n s (\int_0^s 1 \, dt) \, ds = \frac{n^3}{3}$ as $\int_0^n 1 (\int_s^n t \, dt) \, ds = \frac{n^3}{3}$ I would appreciate any pushes in the ...
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1answer
23 views

Cartesian to Spherical Coordinate Conversion for Triple Integral

I have a question regarding what happens to the boundaries when converting a triple integral from Cartesian to Spherical Coordinates. Example ...
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0answers
22 views

Is this Brownian Integral identity correct?

$$\int_0^1 B_t dt=\lim_{\omega \to\infty}{1 \over {\omega}}{\int_0^{\omega}{Y_0+}X_t dt}$$ Where $B_t$ is simple brownian motion, and $X_t$ is a discrete random variable that can be 1 or -1 with ...
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1answer
47 views

A Trig Integral

Does the integral \begin{align} \int_{0}^{\pi/2} \cos(x) \, \ln\left( \frac{1 + a^{2} \sin(x)}{1 - a^{2} \sin(x)} \right) \, dx \end{align} have a closed form and what is changed if the limits are ...
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2answers
28 views

Monotone & bounded of an integral function .

Let , $f: [0,1]\to \mathbb R$ be such that $f(t)\ge 0$ for all $t\in [0,1]$. Define , $$g(x)=\int_0^xf(t) \,dt.$$ Then (A) $g$ is monotone & bounded (B) $g$ is monotone but not bounded (C) $g$ ...
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1answer
32 views

Integrating the log-normal function

Compute $$F(t)=\int_0^t \frac{1}{\sqrt{2\pi}\sigma t} \exp\left[-\frac{1}{2}\left(\frac{\log t-\mu}{\sigma}\right)^2\right]\,dt; t>0$$ My Attempt: $u=\frac{1}{t}\Rightarrow du=-\frac{1}{t^2}dt$ ...
1
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1answer
40 views

Integral involving $\operatorname{sinc}$ and exponential

Is there a closed form for the following integral: $$\int_{0}^a\exp\left[\frac{i\pi x^2}{b}\right]\operatorname{sinc}\left(\frac{\pi ax}{b}\right)dx$$ where $i=\sqrt{-1}$ and ...
2
votes
3answers
115 views

Prove $\int_0^1 ((g'(x))^2-1)^2dx \geq 1$ for smooth $g$ with $g(0)=g(1)=0$ [on hold]

This came up in an optimization problem. How do you prove that $\int_0^1 ((g'(x))^2-1)^2dx \geq 1$ for any $g$ which is twice continuously differentiable on $[0,1]$ and such that $g(0)=g(1)=0$?
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2answers
19 views

double integral question involving volume

find the volume of the solid with height h(x, y)=xy and base D where D is bounded by y=x+2and y=x^2. I believe this is a double integral question. I'm really not sure how to set up the bounds of ...
2
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0answers
26 views

Why do we set $x+1=\frac{1}{t}$ When we compute $I= \int\limits_{\frac{1}{2}}^{0} \dfrac{dx}{(x+1)\sqrt{(3-x)(x+1)}}.$

When we compute $I=\displaystyle \int\limits_{\frac{1}{2}}^{0} \dfrac{dx}{(x+1)\sqrt{(3-x)(x+1)}}.$ We set $x+1=\dfrac{1}{t}$ and we have $\displaystyle I=\int\limits_1^2\dfrac{dt}{\sqrt{4t-1}}$. I ...
2
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1answer
51 views

A definite integral contianing ln(x)

everyone, I met a tough definite integral as follows, $$I = \int\limits_1^\infty {\frac{{\ln x}}{{{{\left( {x + a} \right)}^m}{{\left( {x + b} \right)}^{n + 1}}}}} dx,$$ where $a$ and $b$ are ...
3
votes
2answers
104 views

How to solve:$\int_0^{\infty} \frac{\log(x+\frac{1}{x})}{1+x^2}dx$

Here is my question $$\int_0^{\infty} \frac{\log(x+\frac{1}{x})}{1+x^2}dx$$ I have tried it by substituting $x$ = $\frac{1}{t}$. I got the answer $0$ but the correct answer is $\pi log(2)$. Any ...
0
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0answers
18 views

How to find a sum for power series in a given interval [closed]

for this power series, first thing I tried to do was try to take integrals term by term and then find the definite integral in the given interval this is the integral that I find, but here is my ...