Questions about the evaluation of specific definite integrals.

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3
votes
5answers
45 views

Find $\int_{ - \infty }^{ + \infty } {\frac{1} {1 + {x^4}}} \;{\mathrm{d}}x$

How can we find the integral: $$\int_{ - \infty }^{ + \infty } {\frac{1} {1 + {x^4}}} \;{\mathrm{d}}x$$ I tried to find and got it to be $\cfrac{\pi}{\sqrt2}$. Am I correct? Please help me with an ...
1
vote
0answers
7 views

Question regarding double integrals

Regarding the Buffon's needle case for long needles of length $ l>t, $ (the distance between the parallel lines on the floor), we need to solve the integral $$ \int_{\theta=0}^{\frac{\pi}{2}} ...
0
votes
1answer
28 views

Evluating triple integrals via Spherical coordinates

Use Spherical coordinates to evaluate the triple integral $$\iiint_{\mathrm{x^2+y^2+z^2<z}}\sqrt{x^{2}+y^{2}+z^{2}}\, dV,$$ What I tried Converting $x^2+y^2+z^2<z$ to Spherical coordinates ...
2
votes
0answers
44 views

Integral Contest

Before you answer this OP, please read all the terms and conditions below. Thank you... Today I hold an unofficial little contest on brilliant.org. Now, I will hold it here on Math S.E. It's just for ...
7
votes
0answers
43 views

Real analytic methods for the following integral

A few days back, the following integral was posted $$\int_0^1 x^x(1-x)^{1-x}\sin(\pi x)\,dx=\frac{\pi e}{24}$$ The integral was answered using complex analysis tools but I am interested in other ...
0
votes
0answers
31 views

A Riemann Integrability Question

Define $f:\mathbb{R} \rightarrow \mathbb{R}$. For any fixed closed interval $[a,b] $,$f(x) $ is $Riemann$ integrable on $[a,b].$ Proof:$\forall a,b;c,d\in\mathbb{R},a<b,c<d.$ $f (x+y) $ is ...
0
votes
1answer
12 views

Area between $y=2+|x-1|$ and $y=-\frac{1}{5}x+7$

Question 17, page 448, from Anton 8th. The question asks for the area, and the answer is 24. Now, I did draw the graph, found the points where both functions touch each other by $f(x) = g(x)$: ...
0
votes
0answers
27 views

A inverse Trigonometric multiple Integrals

How to calculate the closed form of the integral $$\int\limits_0^1 {\frac{{\int\limits_0^x {{{\left( {\arctan t} \right)}^2}dt} }}{{x\left( {1 + {x^2}} \right)}}} dx$$
0
votes
1answer
24 views

Is there a clever way to determine negative area of an integral?

Given some continuous, integratable function f(x) that has only positive area over a range from x1 to x2...is there a way to determine the negative area of the integral of f(x) - c (from x1 to x2), ...
8
votes
1answer
77 views

Closed form of $\int_0^1\left(\frac{\arctan x}{x}\right)^n\,dx$

Inspired by this question, is there a closed-form of $$\int_0^1\left(\frac{\arctan x}{x}\right)^n\,dx\,?$$ Here $n \in \mathbb{N_+}$. In the answers to the question above we could find proofs of ...
7
votes
3answers
71 views

Improper integral : $\int_0^{+\infty}\frac{x\sin x}{x^2+1}$

How to evaluate the following improper integral : $$\int_0^{+\infty}\frac{x\sin x}{x^2+1}\,dx$$ I have tried integration by parts and variable substitution, but it didn't work.
3
votes
4answers
144 views

Inverse Trigonometric Integrals

How to calculate the value of the integrals $$\int_0^1\left(\frac{\arctan x}{x}\right)^2\,dx,$$ $$\int_0^1\left(\frac{\arctan x}{x}\right)^3\,dx $$ and $$\int_0^1\frac{\arctan^2 x\ln x}{x}\,dx?$$
3
votes
3answers
81 views

Evaluate integral: $\int_0^{+\infty}\frac{\cos{bx}-\cos{ax}}{x}dx$

It seems that $\displaystyle\int_0^{+\infty}\frac{\cos x}{x}$ is divergent, so how to solve this problem? $$\int_0^\infty \frac{\cos bx -\cos ax}{x}\, dx\quad,\quad\mbox{where}\,a,b>0$$ It's ...
1
vote
1answer
29 views

Change of variable in double integrals

I need help to solve the following question(s). a) Evaulate the integral $$\iint_D (x-y) \, dx \, dy,$$ where $D$ is the triangle with vertices $(0,0)$, $(-1,1)$ och $(4,2)$. b) Evaulate the ...
6
votes
0answers
52 views

Closed form of a difficult definite integral

I'm looking for a closed-form expression for the value of this integral: $$I=\int_0^\pi \frac{\sin(x)}{\sqrt{x^3+x+1}} dx$$ The graph of the integrand looks like this: $\hskip 2.4 in$ Numerically, ...
2
votes
5answers
630 views

How can I show that these integrals are zero

How can I show that these integrals equal $0$ when $n$ and $m$ are both integers and $n \neq m$? $$\int_{-\pi}^{\pi}\sin(mx)\sin(nx)dx = \int_{-\pi}^{\pi}\cos(mx)\cos(nx)dx = 0$$ I'm able to show that ...
7
votes
0answers
38 views

Question on the paper Donal F. Connon, “Some integrals involving the Stieltjes constants”

I'm reading Donal F. Connon, Some integrals involving the Stieltjes constants. It gives a definition of the generalized Stieltjes constants $\gamma_n(u)$ as coefficients in the Laurent series ...
1
vote
3answers
74 views

How to show that these integrals are the same?

While doing some mathematical modelling of planetary orbits I have come up with two definite integrals which appear to produce the same result when I test (using www.WolframAlpha.com) for various ...
9
votes
4answers
141 views

Integral: $\int_0^{\pi/12} \ln(\tan x)\,dx$

I am trying to evaluate: $$\int_0^{\pi/12} \ln(\tan x)\,dx$$ I think the integral is quite simple but I am having a hard time evaluating it. I started with the result: $$\int_0^{\pi/4} \ln(\tan ...
7
votes
2answers
146 views

Evaluating $\int_0^1 \frac{t^{a-1}}{1-t}-\frac{ct^{b-1}}{1-t^c}\ dt$

At first sight it looks like the integral below $$\int_0^1 \frac{t^{a-1}}{1-t}-\frac{ct^{b-1}}{1-t^c}\ dt$$ can be evaluated by using some geometric series. What else can we do? Is there a fast easy ...
1
vote
0answers
22 views

Jacobian determinant of unitary transformation

Is the Jacobian determinant of a unitary transformation equal to one? I ask because I get that impression from the appendix of this paper. They have spherical coordinates for two particles, ...
1
vote
1answer
75 views

A function $f(x)$ that Riemann integrable on $[a,b]$.

Define a function $f(x)$ that Riemann integrable on $[a,b]$. Let $$g(x)=\begin{cases} f(x)&\text{if}&x\in[a,b], \\ f(a)&\text{if}&x<a, \\ f(b)&\text{if}&x>b. ...
3
votes
2answers
59 views

An inequality with $a_n=\int_0^1 \frac{\mathrm{d}x}{\underbrace{\sqrt{2+\sqrt{2+\dots+\sqrt{2x}}}}_{n \text { times}}}$

Let the sequence $(a_n)_n$ defined by $$a_n=\int_0^1 \frac{\mathrm{d}x}{\underbrace{\sqrt{2+\sqrt{2+\dots+\sqrt{2x}}}}_{n \text { times}}}$$ 1)Prove that $$\frac12 \leq a_n \leq ...
3
votes
2answers
69 views

Log Gamma integral

What is the constant $\phi$ in the evaluation \begin{align} \int_{0}^{1/4} \ln\Gamma\left( t + \frac{1}{4}\right) \, dt = \frac{1}{8} \ln(\phi) \end{align} and the constant $\theta$ in the value ...
3
votes
1answer
42 views

$\int_{-\infty}^{+\infty}dx\frac{x\cos(xt)}{e^{ax}-e^{-ax}}$

Apparently from Mathematica we have: $$\int_{-\infty}^{+\infty}dx\frac{x\cos(xt)}{e^{ax}-e^{-ax}}=\frac{\pi^2\mathrm{sech}^2\left(\frac{\pi t}{2a}\right)}{4a^2}$$ for $a,t$ both real and positive. I ...
3
votes
0answers
30 views

Closed form of $\int_0^1 \left(\ln \Gamma(x)\right)^3\,dx$

From the amazing result by Raabe we know that $$LG_1=\int_0^1 \ln \Gamma(x)\,dx = \frac{1}{2}\ln(2\pi) = -\zeta'(0).$$ We also know that $$LG_2 = \int_0^1 \left(\ln \Gamma(x)\right)^2\,dx = ...
6
votes
0answers
61 views
+50

Closed form of $\int_0^1 \operatorname{Li}_3^3(x)\,dx$ and $\int_0^1 \operatorname{Li}_3^4(x)\,dx$

We know a closed-form of the first two powers of the integral of trilogarithm function between $0$ and $1$. From the result here we know that $$I_1=\int_0^1 \operatorname{Li}_3(x)\,dx = ...
6
votes
2answers
46 views

Closed form of $\int_0^1 \operatorname{Li}_p(x) \, dx$

While I've studied integrals involving polylogarithm functions I've observed that $$\int_0^1 \operatorname{Li}_p(x) \, dx \stackrel{?}{=} \sum_{k=2}^p(-1)^{p+k}\zeta(k)+(-1)^{p+1},\tag{1}$$ for any ...
0
votes
1answer
43 views

Calculate a definite integral given the value of another define integral

I'm given that a function $f$ is continuous in $[a, b]$ and given a value $\int_a^b f(u)du = c$. Then I'm asked to calculate $\int_g^h f(t)dt$. I'm looking at the fundamental theorem of calculus but I ...
4
votes
1answer
43 views

Examining the convergence of $\int_{1}^{\infty}\frac{1}{x^2+x}\text{ d}x$

I'd like to have my solution verified for this one. I'd like to show that $$\int\limits_{1}^{\infty}\dfrac{1}{x^2+x}\text{ d}x$$ is convergent. Notice, by partial fraction decomposition, that ...
0
votes
1answer
20 views

asymptotic series for “stable distribution”

I'm trying to understand how to get from one equation to another in a certain paper I am studying (DOI:10.1080/00018738100101467, eqs. 4.34 and 4.35). The equations are pretty self contained, so I'm ...
2
votes
1answer
20 views

Finding Factorial using Integral Definition

$n! = \int_{0}^{\infty} {e}^{-x}{x}^{n} \,dx$ How can we find $400!$? $400! = \int_{0}^{\infty} {e}^{-x}{x}^{400} \,dx$ Integration by parts is way too complicated, what are the other options?
1
vote
1answer
38 views

Find the area bounded by $x+y=3$ and the coordinate axes.

Find the area bounded by $x+y=3$ and the coordinate axes. I know how to find the area bounded by 2 curves it's just that I'm confused with "coordinate axes". Is it the same as x=y? or not? please ...
3
votes
1answer
41 views

Reference for closed form integral of $\int_0^1 dz\,z^n/(z-a)$

Is there a closed form (non-recursive) expression for the definite integral $$\int_0^1 dz \frac{z^n}{z-a}, \qquad n\in\mathbb{Z}_+ \text{ and } a\notin (0,1)$$ for general $n$ and $a$ given in terms ...
5
votes
2answers
72 views

Is this closed form of $\int_0^1 \operatorname{Li}_3^2(x)\,dx$ correct?

According to Freitas' paper at page $11$. $$\int_0^1 \operatorname{Li}_3^2(x)\,dx = 20-8\zeta(2)-10\zeta(3)-\frac{15}{2}\zeta(4)-2\zeta(2)\zeta(3)+\zeta^2(3).$$ I evaluated the LHS and it is ...
5
votes
3answers
83 views

Proving convergence of $ \int \limits_0^{\infty} \cos\left(x^2\right) dx $

How would one prove the convergence of $$ \int_0^{\infty} \cos\left(x^2\right) \,\mathrm dx $$ I tried using the integral test for convergence by noting that making the substitution $u = x^2$ means ...
0
votes
0answers
3 views

find the kernel of voltera 2nd kind with particular form 2. (Alternative approach)

find the kernel of voltera 2nd kind with particular form 2. (Alternative approach) in which we take kernel function of x and t ot just x or just function of t. we try to solve it with alternative ...
2
votes
0answers
29 views

Integral involving logarithm and inverse trigonometric function [on hold]

$${\int\limits_0^1 {\frac{{\ln \left( {1 - x} \right)\ln \left( {1 + x} \right){{\ln }^2}x}}{{1 + x}}} },{\int\limits_0^1 {\frac{{\ln \left( {1 - x} \right)\ln \left( {1 + x} \right){{\ln }^2}x}}{{1 - ...
0
votes
0answers
40 views

help me with this integral $\int_0^{\infty}\frac{1}{x}e^{-kx}\sin xdx=\arctan k^{-1}$ [duplicate]

$\int_0^{\infty}\frac{1}{x}e^{-kx}\sin xdx=\arctan k^{-1}$
8
votes
1answer
123 views

Log integrals IV

It can be determined that the integral \begin{align} \int_{0}^{\pi/2} \frac{x}{\sin(x)} \ln\left(\frac{1+\cos(x) - \sin(x)}{1+\cos(x) + \sin(x)} \right) dx \end{align} has a finite value. Is there an ...
5
votes
1answer
89 views

Integral ${\large\int}_0^1\frac{\ln^2\ln\left(\frac1x\right)}{1+x+x^2}dx$

Gradshteyn & Ryzhik, 7th ed., p. 570, formula 4.325(5) give the following definite integral: ...
0
votes
1answer
43 views

Integral for $\frac{x}{x^2+1}cosx$

When computing Fourier transformation I came across these integral: $$ \int_{\Bbb R}\frac{x \cos x}{1+x^2}\;dx\text{ or } \int_{\Bbb R}\frac{x \sin x}{1+x^2}\;dx $$ Can anyone give me some hints on ...
0
votes
3answers
26 views

Semi Gauss integral limit

I am courrently stuck at showing that: $lim_{x \rightarrow \infty}\int_0^xe^{t^2-x^2}dt=0$. Non of my tries by estimations lead to succes so I would appriciate any kind of help.
0
votes
0answers
11 views

find resolvent kernel of voltera 2nd kind

Find the resolvent kernel of Voltera integral equation with the following kernels $$k(x,t)=29+6t$$ and $$ k(x,t)=5t-6t^2$$
0
votes
2answers
33 views

Substation problem with a simple integral

i have this integral $$ \int {4x\over \sqrt{1+4x^2}} dx $$ and i have tried to solve it by doing like this $$ t=\sqrt{1+4x^2}->t^2=1+4x^2->2tdt=8xdx->tdt=4xdx $$ and im gettin this integral ...
1
vote
0answers
39 views

Measurability and a integral

I need to calculate $\lim_{n\rightarrow\infty}\int^{\infty}_{0}\frac{cos(\frac{x}{n})}{2^x}d\lambda(x)$ and show that the integral makes sense for every $n$. My approach so far: Let ...
1
vote
2answers
34 views

Taking limit with hyperbolic functions

I have a problem with evaluating $$\sinh^{-1}(C \sinh (ax))\bigg|_{-\infty}^{+\infty}$$ where $C$ and $a$ are real positive constants.
8
votes
2answers
137 views

Closed form of $\int_0^1 B_n(x)\psi(x+1)\,dx$

Is there a closed-form of the following integral? $$I_n = \int_0^1 B_n(x)\psi(x+1)\,dx,$$ where $B_n(x)$ are the Bernoulli polynomials and $\psi(x)$ is the digamma function. The motivation of the ...
0
votes
0answers
35 views

Evaluating the antiderivative of a particular improper integral

The task is to integrate $$\tau = \int\limits_{-\infty}^{+\infty}\frac{dx}{\sqrt{E - \frac{U}{cosh^2(ax)}}}, E>U$$ but after taking the integral I get $$\tau ...
3
votes
1answer
114 views

Closed form of $\int_0^{\pi/4} \sin(\sin(x)) \, dx$

Let $I(b)$ is the following integral $$I(b)=\int_0^b \sin(\sin(x)) \, dx.$$ There are some $b$ value for that we know a closed-form of $I(b)$ in term of Struve function $\mathbf{H}_n(x)$. For ...