Questions about the evaluation of specific definite integrals.

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4
votes
1answer
39 views

Improper Intergral (Fresnel- like)

Let $\alpha >1$. Show that $$\int_0^\infty \sin(x^\alpha)\,dx= \sin\left(\frac{\pi}{2\alpha}\right) \int_0^\infty e^{-r^\alpha}\,dr.$$ I was going to ask how to do this but figured it out while ...
1
vote
1answer
22 views

Converting an integrand into a polylog?

Compute the integral $$\int_0^1 dx\,dy\, \frac{\ln(1+y(1-x))}{1-xy}$$ I was just wondering if there is a way to convert the integrand into a polylog? This comes from a tutorial following a lecture ...
0
votes
0answers
11 views

Calculate rotational volumes

I need to calculate the volume from rotating f(x) around y=2x using Pappus–Guldinus theorem. For that I need to know the distance A. $$L = (f(x) - 2x) / 2$$ But how can I optain the distance A?
1
vote
1answer
36 views

How to solve this integral with $\frac{1}{\omega^2}e^\omega$

Please help me solve this integral issue $$\int_{-\infty}^{\infty} \frac{1}{\omega^2} e^{j\omega t} d\omega$$
1
vote
1answer
39 views

How to find bounds of this integral $\int_0^{10} \frac{x}{\sinh \frac{x}{2}}dx$

How to find bounds of this integral: $$\int_0^{10} \frac{x}{\sinh \frac{x}{2}}dx$$ I try but I get that integral not converges. Thank you.
4
votes
3answers
280 views

Decomposition into partial fractions to compute an integral

I'm having problems with: $$\int_{-\infty}^{\infty}\frac{x^4+1}{x^6+1}dx$$ I was thinking: $\frac{x^4+1}{x^6+1}$ is an even function and the interval $(-\infty,\infty)$ is symmetric about 0, we ...
1
vote
0answers
21 views

Covariance of two integrated Brownian motions

I have a question that is similar to the one here: covariance of integral of Brownian, but the answer that I come up with does not match what the book claims the answer is. Given that $$X_t = ...
0
votes
3answers
51 views

Calculate $\lim_{n\rightarrow \infty}\int_{[0,1]}\frac{n\cos(nx)}{1+n^2 x^{\frac{3}{2}}}$

I have tried several methods but even I can not calculate. $$\lim_{n\rightarrow \infty}\int_{[0,1]}\frac{n\cos(nx)}{1+n^2 x^{\frac{3}{2}}}\,dx$$ If anyone can help, it was part of a test and still I ...
6
votes
1answer
80 views
-7
votes
0answers
39 views

What is the integral of $ \int \frac{-6000} { (3x+50)^2} dx$ [on hold]

How can I find the value of the integral $$ \int \frac{-6000} { (3x+50)^2} dx$$
0
votes
0answers
54 views

Evaluate $\large \int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm dx $ using elementary, high school techniques [duplicate]

Evaluate $\large \int_0^1\left(\frac{1}{\ln x} + \frac{1}{1-x}\right)^2 \mathrm dx $ $$$$ I was given this integral by a friend who saw this here on MSE. He asked me if I could solve it using the very ...
-1
votes
1answer
45 views

What is the integral, $\int\frac{dx}{x + \sqrt{1-x²}}\ $? [on hold]

What is the integral, $$\int\frac{dx}{x + \sqrt{1-x²}}\ ?$$
3
votes
0answers
28 views

Definite integral of arcsine over square-root of quadratic

For $a,b\in\mathbb{R}^{+}\land0<a<1$, define $\mathcal{I}{\left(a,b\right)}$ by the integral ...
3
votes
3answers
237 views

double integral $\int_0^t \int_0^s \frac{\min(u,v)}{uv} \, dv \, du$

I want to calculate the double integral: $$\int_0^t \int_0^s \frac{\min(u,v)}{uv} \, dv \, du$$ I don't know how to o that even if it seems simple. Thanks in advance for your help
1
vote
1answer
17 views

Sequence from generating function with integral

So, let $A(x)$ be the generating function of $a_0,a_1,\dots$ then what would be the sequence of the generating function: $$\int^x_0 A(t)dt$$ Since I am not much acquainted with integrals any help ...
0
votes
1answer
15 views

How to find the area bounded by the curves $x=a\cos t$ and $y=b\sin t$ in the first quadrant?

I know the given equations are the parametric equation of an ellipse . The curve meet the x axis at $(a,0)$ in the first quadrant . Now I do this $\int_{0}^{a} y dx$ My book has the following step ...
2
votes
1answer
60 views

integrate this double integral by any method you can. [on hold]

I'm having trouble with this double integral: $$\int_0^2\int_0^{2-x} \exp\left(\frac{x−y}{x+y}\right)\text dy\,\text dx$$
3
votes
1answer
59 views

Integral does not 'converge' despite describing a well-defined area…

I have almost evaluated (where all variables are real including the variable $i$) $$ C_1\int_{a + bt^2}^{i} \frac{r ...
0
votes
0answers
16 views

Definite integral involving Legendre Polynomial

Does anyone know the answer to the following definite integral: $\displaystyle \int_{0}^{\pi}P_{\ell}(\cos\theta)\sin^{k}\theta\, d\theta$ for $k\geq1$, where $P_{\ell}(x)$ is the $\ell$-th Legendre ...
1
vote
0answers
28 views

Integral involving Whittaker function

Consider the following integral: $$ \int_1^{\infty} \frac{e^{u/2}}{u}[-\mathrm{Ei}(-u)]\,W_{1,\imath p}(u)\,du, $$ where $\imath=\sqrt{-1}$ and $p>0$ selected so that $W_{1,\imath p}(1)=0$; here ...
1
vote
1answer
28 views

Evaluating a triple integral

$$\iiint \sqrt{x^2+z^2}\,dx\,dy\,dz$$ I am asked to evaluate this integral over the region $$D:=\left \{ (x,y,z) \in\mathbb{R}^3 :x^2+y^2+z^2 \leq 1 \wedge x^2+y^2+z^2 \leq 2y\right \}$$ I did manage ...
1
vote
1answer
27 views

For what $\alpha$ does the integral absolutely and for what conditionally converge?

For what $\alpha$ does the integral absolutely and for what conditionally converge ? $$\int_{0}^{1}\frac{\ln^{\alpha} (1+x^4)}{x^4}\cos{1 \over x}dx$$ Not sure which criteria to use to prove the ...
2
votes
1answer
53 views

Let $a_n>0$ for $n \geq 1$ and let series: $\sum_{n=1}^{\infty}a_n$ diverge. Let $S_n=a_1+a_2+…+a_n > 1$ for $n \geq 1$

Prove that the series: $$\sum_{n=1}^{\infty}\frac{a_{n+1}}{S_n \ln S_n}$$ diverges and the series : $$\sum_{n=1}^{\infty}\frac{a_{n}}{S_n \ln^2 S_n}$$ converges. (Using the famous criteria I ...
0
votes
0answers
46 views

Calculate integral

I have to calculate $$\operatorname{PV}\int_{-\infty}^{\infty}\frac{1}{\pi}\frac{y}{1+y^2}dy$$ I ended up with ...
1
vote
1answer
82 views

Feynman Integration Problem

$$ I = \frac{\pi^2}{8} - \int_0^1 \frac{\arctan(x)}{\sqrt{1-x^2}} \,dx $$ Evaluate $I$ $$ I = \frac{\pi^2}{8} - \int_0^1 \frac{\arctan(x)}{\sqrt{1-x^2}} \,dx$$ $$f(a) = \int_0^1 ...
1
vote
2answers
36 views

Definite integral calculation

$$\int_0^{2\pi}a^2(1-\cos(x))^2\sqrt{((a-a\cos(x))^2+\sin^2(x))}\,dx$$ After some work I get $$2a^3\int_0^{2\pi}(1-2\cos(x)+\cos^2(x))\sin\left(\frac x2\right)\,dx$$ And stopped:( Any ideas?
3
votes
0answers
69 views

Integral formulas involving continued fractions

Ramanujan posed the following formulas as questions in the Journal of Indian Mathematical Society: $$\int_{0}^{\infty}\dfrac{\sin nx\,\,dx}{{\displaystyle x + \dfrac{1}{x +}\dfrac{2}{x +}\dfrac{3}{x ...
3
votes
1answer
57 views

Sophomore's dream changing “x”

"Sophomore's Dream" says $\sum_{n=1}^{\infty}n^{-n}=\int_0^1x^{-x}$ Can you replace the $x$ and $n$ with $2x$ or $x^3$ (and $2n$ or $n^3$) or something? I would guess not, because replacing $x$ with ...
5
votes
1answer
79 views

Prove $ \lim\limits_{n\to\infty}\int_0^1 f(x)g(nx)\,dx=\int_0^1 f(x)\,dx\int_0^1 g(x) \, dx $ [duplicate]

Let $f$ and $g$ be a real valued continuous functions on $\mathbb{R}$ such that $f(x+1)=f(x)$ and $g(x+1)=g(x)$ for all $x\in \mathbb{R}$. Prove that $$ \lim_{n\to\infty}\int_0^1 ...
1
vote
1answer
49 views

Questions about integration

I'm still a bit confused about definite integration although got the basic idea of how to do integration. The problem is to integrate functions on a uniform distribution over [50, 150]. Firstly ...
3
votes
2answers
99 views

use parseval's identity to evaluate the integral $ \int_{-\pi}^{\pi}\sin^4 xdx$

use parseval's identity to evaluate the integral \begin{equation} \int_{-\pi}^{\pi}(\sin x)^4dx\end{equation} I'm familiar with Parseval's identity which states that for each piecewise continuous ...
4
votes
1answer
129 views

Changing order of integration for the triple integral $ \int\limits_{0}^{2} \int\limits_{0}^{2z} \int\limits_{y}^{2y} f_{(x,y,z)}\; dx\, dy\, dz $

I need to change order of integration for the following triple integral: $$ \int\limits_{0}^{2} \int\limits_{0}^{2z} \int\limits_{y}^{2y} f_{(x,y,z)}\; dx\, dy\, dz $$ The domain of integration is ...
-4
votes
2answers
52 views

Calculate $\int_{0}^{1} \frac{x^3}{\sqrt{1+x^4}}dx$ [closed]

Help with this integral, please! $$\int_{0}^{1} \frac{x^3}{\sqrt{1+x^4}}dx$$
0
votes
4answers
67 views

Estimate from below $\int_0^\pi e^{-t}\cos nt dt$ without calculate it.

Estimate from below the following integral $$\int_0^\pi e^{-t}\cos nt dt$$ without calculate it. Here $n\in\mathbb N$. Any suggestions please?
4
votes
0answers
87 views
+50

Solving integral with spherical bessel functions

I would like to find if possible a solution (closed form) for the following integral: $$\frac{1}{2 \pi}\cdot\int\limits_0^{2\pi}\exp\bigg[-ia(\cos x+\sin x)\bigg]\,j_{0}(b\cos x)\,j_{0}(b\sin ...
1
vote
0answers
38 views

A Combinational identity using permutations

For a distribution {$p_1,p_2, …,p_m$}, with $p_i>0$ and$\sum_1^m{p_i}=1$ , let $J$ be a subset of size $j$, and $m>j\geq1$. It holds that: $$\int_0^1\prod_{i \in J} (x^{-p_i}-1) dx = ...
0
votes
0answers
35 views

Algebra Integral simplification

Let some equation problem final result is like this $0\leq1\leq s\leq t\leq u\leq v$ \begin{align} M=\mathrm{exp}\bigg\{-\pi\lambda v^2+\pi\lambda v^2\bigg(\displaystyle\int_o^s ...
6
votes
2answers
36 views

Definite integral of product of functions

I know it's not correct to write: $$\int_{a}^{b}f(x)g(x) dx = \int_{a}^{b}f(x)dx\int_{a}^{b}g(x)dx$$ This result seems obvious, but I can't think of a way to prove that $\int_{a}^{b}f(x)g(x) dx$ ...
0
votes
1answer
20 views

What happens when the interval of an integral changes from infinity to a constant number?

There exist a calculation about electromagnetic mass: $$m_\mathrm{em} = \int {1\over 2}E^2 \, dV = \int\limits_{r_e}^\infty \frac{1}{2} \left( {q\over 4\pi r^2} \right)^2 4\pi r^2 \, dr = {q^2 \over ...
3
votes
0answers
43 views

Closed form of an infinite series of integrals $\int_{0}^{\eta} \cos nt \cos t \sqrt{\cos^2 t - \cos^2 \eta}$

Let $$ I(n,\eta) = \int_{0}^{\eta} \cos nt \, \cos t \, \sqrt{\cos^2 t - \cos^2 \eta}\; dt $$ where it is known that $0 < \eta \leq \frac \pi 2$. Is it possible to evaluate $S$, the infinite ...
13
votes
3answers
205 views

Integral involving Clausen function ${\large\int}_0^{2\pi}\operatorname{Cl}_2(x)^2\,x^p\,dx$

Consider the Clausen function $\operatorname{Cl}_2(x)$ that can be defined for $0<x<2\pi$ in several equivalent ways: ...
1
vote
1answer
45 views

Changing argument into complex in the integral of Bessel multiplied by cosine

I got a problem solving the equation below: $$ \int_0^a J_0\left(b\sqrt{a^2-x^2}\right)\cosh(cx) dx$$ where $J_0$ is the zeroth order of Bessel function of the first kind. I found the integral ...
0
votes
0answers
50 views

Double Integration Working Help

Help I dont know how to approach this question, I have the answer but dont know how to write a detailed working process of obtaining it. It is supposed to find the surface area of a cone that is $z = ...
2
votes
2answers
63 views

what will be the value of this integral

$$ \large{ \int^{\Large{\frac{\pi}{2}}}_{0} \left[ e^{\ln\left(\cos x \cdot \frac{d(\cos x)}{dx}\right)} \right]dx}$$ We know that $\large{a^{log_a(c)} = c}$. But in this question, the expression in ...
0
votes
1answer
19 views

Establishing a Variant of the Mean Value Property of Harmonic Functions

Let $u:U\to \mathbb{C}$ be harmonic and $\overline{D}(P,r)\subset U$. Verify the following variant of the mean value property of harmonic functions: $$u(P)=\frac{1}{2\pi r}\int_{\partial ...
10
votes
0answers
107 views

The Laplace transform of $\frac{\ln(1+at)}{1+t}$

By expressing the square of the exponential integral as a double integral and then making a change of variables, one can show $$ \int_{0}^{\infty} e^{-2zt} \ \frac{\ln(1+2t)}{1+t} \, dt = \frac{e^{2z} ...
1
vote
1answer
16 views

Limit of Cosine and Sine Fourier Transforms

If I define the cosine and sine Fourier transform as (skipping constant prefactors $(2\pi)^{0.5}$): $$\mathcal{F}_C\{f(x)\}=\int_0^{\infty}\,f(x)\,\cos(\omega x)\,dx$$ and ...
1
vote
1answer
105 views
+50

Functional Analysis, a question that needs clarification.

Find the norm of the linear operator $A:C[-1,1]\to L^p[-1,1]; p\geq1$ that is defined as: $$A(x(t))=\int_{-1}^{1}{{x(s)\over (s-t)^{1 \over 3}}}ds$$ Can someone provide an answer with a little more ...
0
votes
0answers
38 views

MAple 17 won't evaluate my integral

I type this into maple and it won't evaluate it: $$ \int_{0}^{1}\pi ((-y^4+1)^2-(1-y)^2) dy $$ I've also tried $$ evalf(\int_{0}^{1}\pi ((-y^4+1)^2-(1-y)^2) dy)$$ It just returns for both cases $$ ...
0
votes
2answers
43 views

Definite integral from -1 to 0 [closed]

How would I evaluate this definite integral $$ \int_{-1}^{0}\tan x dx- \int_{-1}^{0}\sin^2 x dx $$ All i need to know is what to do when an integral is on an interval of -1 to 0. I could do this ...