Questions about the evaluation of specific definite integrals.
0
votes
1answer
32 views
Derivativation of definite integral
Having:
$f = \int_0^{n}{X_{(t)}dt} + X_{n}$
How can I find:
$\frac{\partial{f}}{\partial{n}} = ?$
Please note that the derivative is done with respect to one of the ends of the integral. (hope ...
9
votes
4answers
271 views
Singular asymptotics of Gaussian integrals with periodic perturbations
At the bottom of page 5 of this paper by Giedrius Alkauskas it is claimed that, for a $1$-periodic continuous function $f$,
$$
\int_{-\infty}^{\infty} f(x) e^{-Ax^2}\,dx = \sqrt{\frac{\pi}{A}} ...
1
vote
1answer
75 views
Evaluating the following integral:
I am trying to evaluate this integral:
$$\int_{0}^{\infty }\frac{\cos(x)}{1+x^{2}}dx$$
My attempt:
$$\int_0^{\infty}\frac{\cos(x)}{(x+i)(x-i)}dx=1/2 \int_{-\infty}^{\infty} ...
2
votes
3answers
84 views
Is it possible to evaluate this integral? [duplicate]
Is it possible to evaluate this integral: $$\int_{0}^{\frac{\pi }{2}}\ln(\sin 2x){\rm d}x$$
4
votes
3answers
146 views
Tricky elementary integral
$$\int_{0}^{\frac{\pi }{2}}x\cot(x)dx$$
I tried integration by parts and got $\frac{1}{2}\int_{0}^{\frac{\pi }{2}}x^{2} \csc^{2}x dx$ which doesn't help at all. I don't really know what to do. Any ...
4
votes
5answers
95 views
A “tricky” integral: $\int_0^{\infty} t e^{-nct} (1-e^{-ct})^m dt$
In an article in the current (May 2013) issue of the College Mathematics Journal,
they say that the following integral is
"tricky to evaluate":
$\int_0^{\infty} t e^{-nct} (1-e^{-ct})^m dt$
where ...
1
vote
2answers
104 views
Not getting $-\frac{\pi}{4}$ for my integral. Help with algebra
Evaluate $$\int_0^\infty \dfrac {\log{x}}{(x^2+1)^2}dx$$
I've been working on this problem for half the day. I'm not getting anywhere.
1) I first changed the integral from negative infinity to ...
0
votes
2answers
64 views
How to calculate the limit $\lim_{n\rightarrow\infty} n\int^{1}_{0}x^{kn}e^{x^{n}}dx$
Let $k$ be a fixed positive integer. How to calculate the following limit? $$\lim_{n\rightarrow\infty} n\int^{1}_{0}x^{kn}e^{x^{n}}dx$$
0
votes
0answers
48 views
Integral of product of normal cdf and pdf
What do you think, is there a closed form solution of the following Integral
$\textbf{ }$
$$\int_{-\infty}^{a-y}n(x)\, N(b-2y-x)\, dx,$$
where
$N(x)=\int_{-\infty}^x n(z)\, dz\quad$ and $\quad ...
3
votes
1answer
71 views
if $f\left(x\right)=\int\limits _{0}^{x}f\left(t\right)dt$ then $f \equiv 0$?
$f:[0,\infty)\rightarrow\mathbb R$ is a function that for all $0\leq a<b\in \mathbb R$, $f_{|[a,b]}:[a,b]\rightarrow \mathbb R$ is integrable.
assuming that for all $x\in[0,\infty)$ , ...
4
votes
1answer
44 views
the definite Integral of $f \cdot g$ when $g$ is integrable and non negative and $f$ is continuous
if $f$ is a Continuous function on $[a,b]$ and $g$ is Riemann integrable and non negative on $[a,b]$ then there exists $\xi\in [a,b]$ and
$$
\int_{a}^{b}f(x)\cdot ...
2
votes
0answers
55 views
Existence of closed form solutions on a definite integral.
I need to find the value of the given expression in a closed form
$$\int_0^{\frac{\pi}{2}} \dfrac{\sin x}{ x} dx$$
My effort:
I know that its a standard definite integral called $Si(x)$
$\sin x= ...
0
votes
3answers
35 views
Confusing Triple Integral
i'm having trouble with this integral
the integral is $\int_0^9\int_{\sqrt z}^3\int_0^y z\cos(y^6)\,dx\,dy\,dz$.
We aren't given any more information and i'm a bit stuck as to where to start. I don't ...
2
votes
2answers
38 views
How to compute the integral $\int_{0}^{\infty} \frac{x^{1/3}}{1+x^{2}} \ dx$ [duplicate]
I want to compute this integral $$\displaystyle\int_{0}^{\infty} \frac{x^{1/3}}{1+x^{2}} \ dx$$
What I did was the following. I substituted $x=t^{6}$, so that my $dx= 6t^{5} \ dt$ and so the integral ...
5
votes
0answers
92 views
Can someone explain this integration trick for log-sine integrals?
I was working on this rather challenging log-sin integral.
$\displaystyle \int_{0}^{2\pi}x^{2}\ln^{2}(2\sin(x/2))dx=\frac{13{\pi}^{5}}{45}$
The upper limit is a waiver from the norm of ...
0
votes
2answers
41 views
Help with algebra involving residues
Evaluate $$\int_0^\infty \dfrac {\cos(mx)}{1+x^4}dx$$
I know I have to change the inegral to $-\infty$ as the lower limit. I understand the logic of the problem but it's the algebra in finding the ...
14
votes
2answers
125 views
Evaluating $\int_0^{\infty} \text{sinc}^m(x) dx$
How do I evaluate $$I_m = \displaystyle \int_0^{\infty} \text{sinc}^m(x) dx,$$ where $m \in \mathbb{Z}^+$?
For $m=1$ and $m=2$, we have the well-known result that this equals $\dfrac{\pi}2$. In ...
1
vote
2answers
53 views
Triple integral problem involving a sphere
Let $R = \{(x,y,z)\in \textbf{R}^3 :x^2+y^2+z^2\le\pi^2\}$
How do I integrate this triple integral
$$\int\int\int_R \cos x\, dxdydz,$$ where $R$ is a sphere of radius $\pi$?
I have trouble ...
1
vote
2answers
37 views
Does $\lim\limits_{n\to\infty} \sum\limits_{r=0}^{\lfloor\frac{n}{2}\rfloor}\frac{1}{n}f(\frac{r}{n})$ imply $\int\limits_{0}^{\frac{1}{2}}f(x)dx$? [duplicate]
We know that $\lim\limits_{n\to\infty} \sum\limits_{r=0}^{n-1}\frac{1}{n}f(\frac{r}{n})$ implies $\int\limits_{0}^{1}f(x)dx$.
Then does $\lim\limits_{n\to\infty} ...
5
votes
2answers
121 views
Trig Fresnel Integral
$$\int_{0}^{\infty }\sin(x^{2})dx$$
I'm confused with this integral because the square is on the x, not the whole function. How can I integrate it? Thank you.
I have not done complex analysis (only ...
2
votes
0answers
43 views
Evaluate $\int_0^\tau \frac{t\sin(t z)}{z\cos(t z)-\sin(tz)}\text{d}t$
I'm trying to evaluate the following definite integral. Mathematica gives me a complicated expression which I think I can simplify, but I was wondering if there was a "nice" way to evaluate it.
...
2
votes
1answer
80 views
What is the proof that anti-derivative gives function = area under curve?
For many years now I have thought about this but have not been able to get a clear answer. We all know that $\displaystyle \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$ gives us a function we call as the ...
1
vote
2answers
70 views
Find the derivative of y with respect to x,t,or theta, as appropriate
$$y=\int_{\sqrt{x}}^{\sqrt{4x}}\ln(t^2)\,dt$$
I'm having trouble getting started with this, thanks for any help.
3
votes
3answers
62 views
Path independence of an integral?
I'm studying for a test (that's why I've been asking so much today,) and one of the questions is about saying if an integral is path independent and then solving for it.
I was reading online about ...
1
vote
2answers
28 views
Is it true that $\int_1^ba^{\log_b x}dx> \log_eb$
Is it true that
$\int_1^ba^{\log_b x}dx> \log_eb$
$\forall a,b>0\ and\ b\not = 1$
0
votes
2answers
32 views
Determine the area of the region bounded by $y=2e^x$, $ y=e^{2x}$ and $x=0$
$$y_1 = 2e^x$$
$$y_2 = e^{2x}$$
$$x=0$$
I was thinking of finding the $x$-intercepts first, so $2e^x= e^{2x}$.
What is next?
1
vote
1answer
25 views
definite integral negative variable
Man, it's been so long since I did this. I am trying to do this:
NB: limits are $-\pi$ and 0, but I can't get the minus in the limits. If anybody knows how do to that please let me know, the $\pi$ ...
1
vote
3answers
86 views
Is this definite integral impossible?
From my understanding when you integrate $f(x)$ you get $F(x)+C$, and when finding a definite integral the $C's$ cancels out due to subtraction. However, I came across an example where the $C$ doesn't ...
6
votes
1answer
77 views
integration as limit of a sum
If $f$ is continuous on $[0, 1]$ then
$$\lim_ {n\to\infty}\sum_{j=0}^{\lfloor n/2\rfloor} \frac1{n}f\left(\frac {j}{n}\right) = ? $$
will the answer be that the limit exists and is equal to $ ...
0
votes
2answers
50 views
Definite integrals and integral rules
Suppose $\int^{-1.5}_{-6}f(x)$dx = 1, $\int^{-4.5}_{-6}f(x)$dx =9, $\int^{-1.5}_{-3}f(x)$dx = 9.
I solved $\int^{-3}_{-4.5}f(x)$dx = -13. However, how do I get $\int^{-4.5}_{-3}(1(f(x))-9)$dx? I know ...
0
votes
2answers
46 views
Proving an integral equation
for the following Question, i had to prove this :
that for every $$-1\le y \le 1 \\
\arcsin(y) + \arccos(y) = \frac{\pi}{2}$$
NOTE: this I've shown this using basic trigonometric id's
and ...
1
vote
0answers
26 views
Complex Fourier series of a function [duplicate]
I need to find the complex Fourier series of this function, and I'm having problems calculating these integers:
$$|a|<1$$
$$x\in [-\pi,\pi]$$
$$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$
...
2
votes
2answers
181 views
Complex Fourier series
I need to find the complex Fourier series of this function, and I'm having problems calculating these integers:
$$|a|<1$$
$$x\in [-\pi,\pi]$$
$$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$
...
1
vote
1answer
95 views
Proof that Riemann-integrability is preserved by changing the function at a converging sequence of points
Let $(x_n)^{\infty}_{n=1}\in[a,b]$ such that $\lim\limits_{n\to\infty}x_n=l$. Let $g:[a,b]\to \mathbb{R}$ be bounded function. Assume that there exists a Riemann integrable function $f:[a,b]\to ...
14
votes
1answer
155 views
How to prove $\frac{\pi^2}{6}\le \int_0^{\infty} \sin(x^{\log x}) \ \mathrm dx $?
I want to prove the inequality
$$\frac{\pi^2}{6}\le \int_0^{\infty} \sin(x^{\log x}) \ \mathrm dx $$
There are some obstacles I face: the indefinite integral cannot be expressed in terms of ...
1
vote
0answers
16 views
Darboux integrable periodic function is integrable on any closed and bounded interval
How do I show that a Darboux integrable periodic function, with period $T$, is integrable on any closed and bounded interval?
Intuitively it is so clear to me, but how one can provide a proof?
...
5
votes
1answer
129 views
Snags when discovering the asymptotic behavior of an integral
I have trouble in discovering the asymptotic behavior (i.e, the asymptotic expansion) of the following integral:
$$\newcommand\abs[1]{\left\lvert#1\right\rvert}
\int_0^{\pi/2}\frac{dx}{1+(n\pi+x)\sin ...
1
vote
3answers
95 views
Is this integral right?
$$\pi\int_0^{x}\left(\cot(\pi t)-\frac{1}{\pi t}\right)dt=\log\frac{\sin(\pi x)}{\pi x}$$
(original image)
Is this integral right? Regardless of whether it's right or not, please give me a procedure ...
1
vote
1answer
38 views
Simplifying a double integral
How can I evaluate this? Is there a trick or simplification that will make it nicer?
$$ \frac{1}{4\pi} \frac{\partial}{\partial t}\left(2t\int_0^{2\pi}\int_0^1{(x+tr\cos\theta)^2 ...
2
votes
2answers
53 views
Darboux's sums of $f(x)=\frac{1}{x}$
Consider the function $f(x)=\frac{1}{x}$ on $[1,2]$.
How to find $\epsilon>0$, s.t for all partition $P$ of $[1,2]$ with $\lambda(P)<\epsilon$, we will have that
$$|U_{f,P}-L_{f,P}|<0.01$$
...
7
votes
2answers
86 views
Evaluating $\int_{\mathbb{R}}\frac{\exp(-x^2)}{1+x^2}\,\mathrm{d}x$
I would like to evaluate in a closed form the integral
$$\int_{\mathbb{R}}\frac{\exp(-x^2)}{1+x^2}\,\mathrm{d}x$$
I tried various methods :
integration by parts
some changes of variables ...
0
votes
3answers
51 views
General result of the following integral
Let $I = \int\limits_0^\infty dx x^{2n} e^{-\alpha x^2}$
I tried it as follows:
with the substitution $y = x^2 \implies \frac{dy}{dx} = 2x \implies dx = \frac{dy}{2x}$ the integral transforms into ...
2
votes
1answer
29 views
Correct way to calculate a complex integral?
I have
$$
\int_{[-i,i]} \sin(z)\,dz
$$
Parametrizing the segment $[-i,i]$ I have, if $t\in[0,1]$
$$
z(t) = it + (1-t)(-i) = 2it-i, \quad \dot{z}(t) = 2i.
$$
So
$$
\int_{[-i,i]} \sin(z)\,dz = \int_0^1 ...
3
votes
1answer
24 views
Using a particular image to justify a (specific) trig integral equality.
I would like to include the following string of equalities in a paper:
$$\sin ^2(x) + \cos ^2(x) = 1$$
$$\int _0^{\dfrac{\pi}{2}} \sin ^2 (x)dx + \int_0^{\dfrac{\pi}{2}} \cos ^2 (x)dx = ...
0
votes
1answer
54 views
Prove that $\int_0^x \int_0^y \int_0^z f(t) dt dz dy = \frac{1}{2} \int_0^x (x-t)^2 f(t) dt$
Prove that
$$\int_0^x \int_0^y \int_0^z f(t) dt dz dy = \frac{1}{2} \int_0^x (x-t)^2 f(t) dt$$
Came across this problem and I'm not even sure how to start it. I figured that if the end goal is ...
0
votes
1answer
24 views
Finding a horizontal line of intersection of a function under the constraint that the area between the intersection points equals a specified value.
I'm not quite sure what kind of problem this is called. I guess it may be some kind of solving of a system of nonlinear equations. My math is a little rusty.
Suppose I have a function $f(x)$ like the ...
2
votes
1answer
36 views
Evaluation of an integral involving hyperbolic sine and exponential
I am wondering if the following integral can be reduced to either a closed form involving elementary functions, or well-known special functions (such as $\operatorname{erf}$, Bessel functions, etc.):
...
0
votes
2answers
74 views
Calculating the integral $ \int_{0}^{5} { \frac{|x-1|}{|x-2| + |x-4|} } dx$
How do we calculate the following integral:
$$ \int_{0}^{5} { \frac{|x-1|}{|x-2| + |x-4|} } dx$$
3
votes
0answers
20 views
fancy about some properties of kernel functions at infinity
Consider the two common types of kernel functions $\sum\limits_{t=a}^bf(t)K(x,t)$ and $\int_a^bf(t)K(x,t)~dt$ , prove whether the following properties are correct or not:
$1.$ If $K(x,t)$ is bounded ...
1
vote
2answers
101 views
Evaluating a double integral involving exponential of trigonometric functions
I am having trouble evaluating the following double integral:
$$\int\limits_0^\pi\int\limits_0^{2\pi}\exp\left[a\sin\theta\cos\psi+b\sin\theta\sin\psi+c\cos\theta\right]\sin\theta d\theta\, d\psi$$
...
