Questions tagged [definite-integrals]
Questions about the evaluation of specific definite integrals.
12,995 questions
0
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8 views
Definite integral $\int_0^1\Bigl| e^x-a\Bigl|dx, a>0$ and its smallest value
How can I evaluate the smallest value of $I_{(a)}=\int_0^1\Bigl| e^x-a\Bigl|dx, a>0$ and express it like this: $\Bigl(\sqrt{e}-E\Bigl)^F$, where the smallest value of I(a) is took at $a=e^{\frac{G}{...
0
votes
2answers
25 views
Calculate $\iint_D\frac{a\sqrt{2az-z^2}}{z}\text{d}y\text{d}z$
I'm trying to integrate this:
$$
\iint_D\frac{a\sqrt{2az-z^2}}{z}\text{d}y\text{d}z
$$
on
$$
D=\begin{cases}
2z^2-2az=y^2\\
a\le z\le 2a
\end{cases}
$$
But after trying some substitution, I can't find ...
0
votes
0answers
4 views
Solve integral of $f(\lambda)=\frac{\lambda^{\alpha+\beta}}{\prod_{s=1}^n (1/U_s+\lambda)^{\alpha+1}\prod_{s=1}^n (1/T_s+\lambda)^{\beta+1}}$
I need to compute the numerical value of this integral, hundred thousand of times, for a typical dataset. How can I get a good approximation.
$$\int_0^\infty f(\lambda) d\lambda $$
$$f(\lambda)=\frac{...
0
votes
1answer
26 views
Fourier transform of a Gaussian wave packet in position representation to momentum representation
I am Studying Gaussian wave packets in J.J Sakurai's Quantum Mechanics, I stuck here , I don't know to evaluate this integral
$\int_{\mathbf -\infty}^\infty\exp\Biggl({\mathbf -ipx\over h }+ikx-{x^2\...
6
votes
3answers
178 views
How to evaluate the following integral involving a gaussian?
I want to evaluate the following integral:
$$\int\limits_0 ^\infty {x \sin{px} \exp{(-a^2x^2})} dx$$
Now I am unsure how to proceed. I know that this is an even function so I can extend the limit ...
2
votes
1answer
48 views
Is integral of a function differentiable? [on hold]
If we have a continuous function $f(x)$ and its integral is $F(x)=\displaystyle \int_a^x f(x)\ dx$, will $F(x)$ be differentiable?
3
votes
0answers
34 views
Trigonometric integral related to Gieseking's constant
This question at MathOverflow https://mathoverflow.net/questions/302982/how-to-prove-the-identity-l2-frac-cdot3-frac215-sum-limits-k-1-inf conjectures certain relation between fast converging ...
4
votes
2answers
110 views
using Leibniz rule to solve definite integral
The total number of distinct $x$ in $[0,1]$ for which $$\int_{0}^{x}\frac{t^{2}}{1+t^{4}}dt=2x-1$$ holds is
I tried solving using Leibniz rule but that gives $0$ no. of solutions. However the answer ...
0
votes
1answer
26 views
How to determine the radius when utilizing the disc method
Lately, I have been struggling to figure out how to determine the radius used in the formula $V=\pi r^2$ when finding the volume of a solid revolving around an axis. From my experience with solving ...
1
vote
1answer
39 views
Integral with log of absolute value of sine
Show that
$$\int_{-\pi/3}^{\pi/3} \log \vert 8 \sin(t/2) (1 + \sin t)^2 \vert dt = 0.$$
WolframAlpha claims that this is true. I've tried manipulating the integrand a bunch and various trig ...
1
vote
2answers
72 views
Suppose that the average value on all intervals $[a,b]$ is equal to $f((a+b)/2)$. Prove that $f''(x) = 0$ for all $x \in \mathbb{R}$
I understand that $f(x)$ must be linear with a first derivative equal to a constant. I'm just not sure how I can use the mean value property of integrals to show something about $f''(x)$. The hint on ...
1
vote
0answers
73 views
Tough integral from particle physics
I have been struggling to check if it's even possible to calculate the following integral
$$
\iiint_{\Bbb R^3} d^3 \textbf{q} {1\over \sqrt{E_{\textbf{p}+\textbf{q}}(E_{\textbf{p}+\textbf{q}}+m)}}e^{-...
0
votes
2answers
49 views
Proving Integrals
How to prove that
$$ \int_{0}^{\frac{\pi}{2}} \sqrt{\tan x} \ dx = \int_{0}^{\frac{\pi}{2}} \sqrt{\cot x} \ dx$$
I have no idea what I'm gonna do first. I just checked the graph, and the result is the ...
1
vote
0answers
25 views
How to derive the closed form for $\int_0^1 \left(\ln(\Gamma(z))\right)^2dz$
Is there any way to derive this identity?$$\int_0^1 \left(\ln(\Gamma(z))\right)^2dz=\frac{\gamma^2}{12}+\frac{\pi^2}{48}+\frac{\gamma\ln(2\pi)}{6}+\frac{(\ln(2\pi))^2}{3}-(\gamma+\ln(2\pi))\frac{\zeta'...
0
votes
0answers
26 views
Number of solutions within an interval
What I'm trying to calculate is the following $$\int_0^{2\pi}dz_1\sum_{j\text{ s.t. } z_2^{(j)}=\arccos(\lambda-\cos(z1))\text{ or }\\\text{ written another way}\\0\leq\arccos(\lambda-\cos(z1))\leq\...
0
votes
1answer
45 views
Prove $\sqrt{1-e^{-1}}<\frac{1}{\sqrt{\pi}}\int^1_0 e^{-x^2}dx<\sqrt{1-e^{-2}}.$
Problem
Prove $$\sqrt{1-e^{-1}}<\frac{1}{\sqrt{\pi}}\int^1_0 e^{-x^2}dx<\sqrt{1-e^{-2}}.$$
Attempt
Since $$ \forall x\in[0,1],\quad e^{-x^2}=\sum_{n\geq 0}\frac{(-1)^n}{n!}\,x^{2n}$$
and by ...
0
votes
1answer
40 views
Is there a way to mathematically prove $\psi (\mathbf{r})$ varies continuously (using the intuitive arguments provided below)?
Electric potential at a point outside the charge distribution is:
$\displaystyle \psi (\mathbf{r})= \int_{V'}
\dfrac{\rho (\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|} dV'$
where:
$\mathbf{r}=(x,y,z)$ ...
4
votes
1answer
112 views
Integrals of the form $\int \log(2+2\cos x)^ndx$
$\log$ will be the natural logarithm and $\zeta$ the Riemann zeta function. I'm interested in the following family of integrals:
$$
I_n = \int_0^\pi(\log(2+2\cos x))^n\mathrm{d}x
$$
Some of the values ...
0
votes
3answers
56 views
If $I_{n}=\int^{1}_{0}x^2(1-x^2)^ndx,$ Then $\lim_{n\rightarrow \infty}\frac{I_{n+1}}{I_{n}}$ is
If $\displaystyle I_{n}=\int^{1}_{0}x^2(1-x^2)^ndx,$ Then $\displaystyle \lim_{n\rightarrow \infty}\frac{I_{n+1}}{I_{n}}$ is
plan
$$I_{n}=-x\cdot \frac{(1-x^2)^{n+1}}{2(n+1)}\bigg|^{1}_{0}+\frac{...
3
votes
2answers
59 views
Remove even elements of partition of integration set
Suppose I am integrating a continuous $f:\mathbb{R}\rightarrow\mathbb{R}$
in a measurable set $A\subseteq I$, where $I$ is an interval:
$$
\int_{A}f(x)dx
$$
Now suppose I partition the set $I$ in $N$ ...
0
votes
0answers
54 views
Tricky integral limit [duplicate]
Please help me to solve the following limit using high-school methods(no approximations or DCT). I was thinking to use the squeeze theorem but I can't find the best boundaries to evaluate it.
$$\lim_{...
0
votes
1answer
41 views
How was this integral with functions as limits solved? [duplicate]
How was this integral solved ?
$$\displaystyle \frac d{dy}\int_{g(y)}^{h(y)}f(x,y)dx$$ $$=\int_{g(y)}^{h(y)}\frac \partial{\partial y} f(x,y)dx+f(h(y),y)\frac{dh(y)}{dy}-f(g(y),y)\frac{dg(y)}{dy}$$
...
1
vote
1answer
70 views
Evaluation of integral $\int_{-\infty}^{\infty} e^{itx} \frac{1- e^{-\frac 1 2 t^2}}{\frac 1 2 t^2} \text d t$ / specific characteristic Function
I want to calculate the value of
$$I(x) :=\int_{-\infty}^{\infty} e^{itx} \frac{1- e^{-\frac 1 2 t^2}}{\frac 1 2 t^2} \text d t$$
where $x\in \Bbb R$. Of course we can write
$$I(x) = \int_{-\infty}^{\...
1
vote
2answers
61 views
Even stronger than Sophomore's dream [duplicate]
Sophomore's dream states that:
$$
\int_0^1x^{-x}dx=\sum_{n=1}^\infty n^{-n}
$$
and
$$
\int_0^1x^{x}dx=-\sum_{n=1}^\infty(-n)^{-n}
$$
A friend of mine noticed that numerically:
$$
\int_0^1\int_0^1(xy)^{...
0
votes
2answers
32 views
How to explain or give reasons about the curve?
I know that arc length of the curve is:
$$s=\int_{a}^{b}\sqrt{1+\big(f^{'}(x)\big)^2}dx$$
The question is
"Is there a smooth curve y=f(x)" where length over the interval $0\leq x \leq a$, (where a is ...
0
votes
1answer
32 views
How to find the length of the curve?
I know that length of the curve is either of:
$$
s = \int_a^b \sqrt{1+\left(\frac{dy}{dx}\right)^2}\ dx
= \int_c^d \sqrt{1+\left(\frac{dx}{dy}\right)^2}\ dy$$
Now the curve is $y=x^2$, $-1\leq x\...
1
vote
2answers
43 views
integration of a gaussian with $x^2$
I need to integrate
$$\int_{-\infty}^{\infty} x^2 e^{-ax^2} \qquad \text{where } a\in R$$
The book does the following:
I don't understand what's happening. I tried solving the integral using ...
1
vote
1answer
98 views
Closed-forms for the integral $\int_0^1\frac{\rm{Li}_n(x)}{1+x}dx$?
(This is related to this question.)
Define the integral,
$$I_n = \int_0^1\frac{\rm{Li}_n(x)}{1+x}dx$$
with polylogarithm $\rm{Li}_n(x)$. Given the Nielsen generalized polylogarithm $S_{n,p}(z)$,
$$...
5
votes
0answers
143 views
More on the integral $\int_0^1\int_0^1\int_0^1\int_0^1\frac{1}{(1+x) (1+y) (1+z)(1+w) (1+ x y z w)} \ dx \ dy \ dz \ dw$
In this post, the OP asks about the integral,
$$I = \int_0^1\int_0^1\int_0^1\int_0^1\frac{1}{(1+x) (1+y) (1+z)(1+w) (1+ x y z w)} \ dx \ dy \ dz \ dw$$
I. User DavidH gave a beautiful (albeit long)...
2
votes
2answers
119 views
Hard and weird Integration in numerator and denominator
$$
\frac{\int_0^\pi x^3\ln(\sin(x))\ dx}
{\int_0^\pi x^2\ln(\sqrt{2}(\sin(x))\ dx}$$
In this problem , I'm unable to understand how to start. I tried applying By parts but I couldn't solve it . ...
4
votes
0answers
35 views
Convolution of a box function over the hypercube
This looks like a simple convolution over the $p$-dimensional hypercube, but I am unable to find a closed form expression for arbitrary integer $p$:
$$f_p(a)=\int_0^1 dx_1\int_0^1 dx_2\cdots \int_0^1 ...
0
votes
0answers
38 views
Evaluate the definite integral $\int_0^{\infty} \frac{x}{e^x-e^{-x}}$ [duplicate]
Evaluate the definite integral $$\int_0^{\infty} \frac{x}{e^x-e^{-x}}$$
I couldn't get the system at first that is why I deleted my first post. I couldn't find how to edit it so I decided to ask here....
0
votes
1answer
70 views
Find the value of $\int_0^1xf(x)dx$
Given:
$f(0)=0, f(1)=2, \text{ and} \int_0^1f(x)dx=3$
Find the value of $\int_0^1xf(x)dx$
Attempt:
Using partial integration.
$$
\int xf(x)dx=\frac{x^2f(x)}{2}-\int\frac{x^2f'(x)}{2}dx
$$
Maybe I'...
0
votes
2answers
41 views
Prove $|\int_{n}^{n+p} \sin (x^2)dx|\leq 1/n$ where $p>0$.
Prove $$\left|\int_{n}^{n+p} \sin (x^2)dx\right|\leq \frac{1}{n}$$ where $p>0$.
Maybe, we can improve namely enhance to prove $$\left|\int_{n}^{+\infty} \sin (x^2)dx\right|\leq \frac{1}{n},$$
...
0
votes
0answers
35 views
Finding Limit involving definite integral [closed]
Suppose the limit
$$L = \lim_{n\to\infty} \sqrt{n} \int_{0}^1 \frac{1}{(1+x^2)^n}dx$$
exists and is larger than $\frac{1}{2}$.
Then which of the following inequalities are true?
A) $...
0
votes
1answer
27 views
Evaluating $\int_{-\pi}^{\pi} dq \; e^{iqn} e^{i \alpha\cos(q)}$
I am trying to solve this definite integral
$$\int_{-\pi}^{\pi} dq \; e^{iqn} e^{i \alpha\cos(q)}$$
where $\alpha \in\mathbb{R} $ and $n\in\mathbb{Z}$.
I know that for $n=0$
$$\int_{-\pi}^...
0
votes
0answers
21 views
How to set up a line integral of a vector field over the borders of two circumferences
So...I have found a problem where I have to solve the integral of the vector field:
$F(x,y)=(\sin(x)ln(x)+y^2) a_x + (\cos(y)e^y-x^2) a_y$
Along the borders of the region bounded by the ...
2
votes
2answers
54 views
Is $\int_0^1 \frac{x^n}{\sqrt{1-x^4}}dx$ convergent?
$$\int_0^1 \frac{x^n}{\sqrt{1-x^4}}dx$$
Near $0$ the expression inside is convergent, that is easy. Near $1$ looks like it approaches infinity when $n \ge 0$
But according to the book when $n \ge -1$ ...
1
vote
1answer
70 views
How can I solve this kind of integral? [closed]
Evaluate
$$\int_{-\infty}^{\infty}\frac{1}{z}e^{-i(Az^2 - Bz)}dz,$$ where A and B are some coefficients.
1
vote
2answers
33 views
Prove $\ln \int_0^1 f(x)dx \geq \int_0^1 \ln f(x) dx$.
Let $f(x) \in C[0,1]$, and $f(x)>0$ over $[0,1]$. Prove $$\ln \int_0^1 f(x)dx \geq \int_0^1 \ln f(x) dx.$$
If we denote
$$F(x):=\ln \int_0^x f(t){\rm d}t-\int_0^x \ln f(t){\rm d}t, ~~~x \in[0,1]$$ ...
5
votes
2answers
63 views
Let $f:[a,\infty)\rightarrow \mathbb{R}$ be a uniformly continuous function. $\int_{a}^{\infty} f$ converges.Prove that $\lim_{x\to\infty} f(x)=0$
Let $f:[a,\infty)\rightarrow \mathbb{R}$ be a uniformly continuous function in that range. $\int_{a}^{\infty} f$ converges. Prove that $\lim_{x\to\infty} f(x)=0$
Hint: Use the sequence $F_n(x)=n\...
1
vote
1answer
48 views
Differentiation when there are continuously many variables
Suppose there are a continuum of tasks in a unit range $[0, 1]$, and for each task $i \in [0, 1]$, a firm can choose the amount of robots, $x_i$. I am hoping to get the first-order condition (e.g., $\...
3
votes
0answers
52 views
Can $1/\log(2)$ be represented as a period?
In this article by Zagier-Kontsevich, period is defined as values of integral of a rational function over a domain in $\mathbb{R}^{n}$ defined by polynomial inequalities with rational coefficients. ...
0
votes
0answers
61 views
Evaluate $\int_0^1 \frac{\ln(1-x^2)\arcsin^2 x}{x^2} {\rm d}x$ [duplicate]
Evaluate $$I=\int_0^1 \frac{\ln(1-x^2)\arcsin^2 x}{x^2} {\rm d}x.$$
Maybe, we can make a substitution $x=:\sin u, u\in [0,\pi/2]$. Then
$$I=2\int_0^{{\pi}/2}\frac{u^2\cos u\ln\cos u}{\sin^2 u}{\rm d}...
6
votes
3answers
103 views
Can't solve $\int_{0}^{\pi} \frac{x}{1 + \cos^2x} dx$
I tried this :-
Let $$I =\int_{0}^{\pi}\frac{x}{1 + \cos^2x}dx\tag{1}$$
then $$I = \int_{0}^{\pi}\frac{\pi-x}{1 + \cos^2(\pi-x)}dx= \int_{0}^{\pi}\frac{\pi-x}{1 + \cos^2x}dx\tag{2}$$
Adding (1) and (...
0
votes
3answers
41 views
two dimensional integral of delta function
For $x,y \in \mathbb{R}$, function $f(x,y)$ is defined as
$$f(x,y) = 1 \quad\textrm{if}\quad x=y$$
$$f(x,y) = 0 \quad\textrm{if}\quad x\neq y$$
It seems to me that the integral $I = \int_0^1 \int_0^1 ...
-2
votes
3answers
86 views
Problem with integrating $\int_0^{\pi/2}\frac{\cos^6x}{\cos^6x+\sin^6x}dx$ [duplicate]
Someone told me there is an equation
$$\int_0^{\pi/2}f(sinx)dx=\int_0^{\pi/2}f(cosx)dx$$
With this equation, it's easy to get the answer$\frac{\pi}{4}$.
What I want to know is why we have this ...
0
votes
1answer
28 views
Help with Contour Integration with Finite Limits
I need to perform this integral:
$$
\int_0^{\omega_{\large s}/\left(4\pi\right)}
\frac{\mathrm{i}\omega\,\mathrm{e}^{\mathrm{i}\omega t}}
{\left(\mathrm{i}\omega - \omega_{0}\right)
\left(\mathrm{i}\...
0
votes
1answer
34 views
Proving the accuracy for numerical integration [on hold]
Given a smooth function $f$, we denote $L_{f}$ the Lagrange polynomial
of degree less than or equal to $1$ which is equal to $f$ at the
points $x_1$ and $x_2$. Define $I_{f} = \int_{-1}^{1} L_{f}(...
1
vote
0answers
35 views
What are the similarities between these 3 limits?
I have 3 limits and I'm pretty sure that the process of solving them it's the same.
$L_1=\lim_{n\rightarrow \infty }n\cdot \int_{0}^{1}x^{2n}\sin(\frac{x\cdot \pi)}{2})dx$
$L_2=\lim_{n\rightarrow \...
