Tagged Questions
-7
votes
0answers
61 views
Why Riemann integration is needed? [closed]
What is the necessity of the notion of "Riemann Integration" ? Why is normal definite integral is not good enough ?
0
votes
0answers
51 views
Integral involving exponential, power and Bessel function
Is there any formula for calculating the following definite integral, including exponential and Bessel function?
$$ \int_0^{a}x^{-1} e^{x}I_2(bx)dx$$
Thanks in advance
3
votes
2answers
58 views
Explanation for these transformations of integrals
I've recently found the following transformations:
$$\int _{a} ^{\infty} \frac{\ln x}{x^2 + a^2}\,dx = \int _{a} ^{0} \frac{\ln x}{x^2 + a^2}\,dx$$
$$\int _{0} ^{\pi/2} ...
4
votes
2answers
79 views
Find the following integral (most likely substitution)
$$\int_0^1 \frac{\ln(1+x^2)}{1+x^2} \ dx$$
I tried letting $x^2=\tan \theta$ but it didn't work. What should I do?
Please don't give full solution, just a hint and I will continue.
5
votes
2answers
69 views
Definite trig integral [duplicate]
How do I evaluate:
$$\int_{0}^{\pi} \sin (\sin x) \ dx$$
I have seen a similar question here but can't find it.
4
votes
3answers
94 views
How to prove that $\lim\limits_{n\to\infty}\int\limits _{a}^{b}\sin\left(nt\right)f\left(t\right)dt=0\text { ? }$
Let $f:\left[a,b\right]\to\mathbb{R}$ be a function that is derivative so that $f'$ is continuous then
$$
\lim_{n\to\infty}\int\limits _{a}^{b}\sin\left(nt\right)f\left(t\right)dt=0
$$
My attempt: I ...
1
vote
3answers
81 views
Integrating a school homework question.
Show that $$\int_0^1\frac{4x-5}{\sqrt{3+2x-x^2}}dx = \frac{a\sqrt{3}+b-\pi}{6},$$ where $a$ and $b$ are constants to be found.
Answer is: $$\frac{24\sqrt3-48-\pi}{6}$$
Thank you in advance!
1
vote
1answer
75 views
Evaluating the following integral:
I am trying to evaluate this integral:
$$\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 ...
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 ...
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 ...
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 ...
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
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
28 views
Is it true that $\int_1^ba^{\log_b x}dx> \log_eb$
Is it true that
$\int_1^ba^{\log_b x}dx> \log_eb$
$\forall a,b>0\ and\ b\not = 1$
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 ...
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
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 ...
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
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 ...
4
votes
2answers
50 views
Integration solving problem
A integration is given $$x-x_0 = \pm \int_{0}^{\phi(x)}\frac{d\Phi}{\sqrt\frac{\lambda}{2}(\Phi^2-\frac{m^2}{\lambda})} \tag{1}$$
The author said that, equation (2) can be written from equation (1) by ...
13
votes
3answers
235 views
integral with $\log\left(\frac{x+1}{x-1}\right)$
I encountered a tough integral and I am wondering if anyone has any ideas on how to evaluate it.
$$\displaystyle ...
7
votes
2answers
161 views
Let $f:[a,b]\to\mathbb R$ be Riemann integrable and $f>0$. Prove that $\int_a^bf>0$. (No Measure theory) [closed]
Is the Riemann integral of a strictly positive function positive?
This is not a duplicate. I'm specifically interested in a proof not involving Measure Theory. The thread above uses the fact that $f$ ...
0
votes
0answers
105 views
Riemann sums vs Darboux sums
Let we speak of the tagged partitions of an interval and a bounded function defined on it.
I think that the tags give rise to particular Riemann sums which may be quite different in value that the ...
4
votes
3answers
77 views
Integral with Bessel functions of the First Kind.
I'd like to solve the following integral:
$I = \int_0^\infty J_0(at) J_1(bt) e^{-t} dt\ $
where $J_n$ is an $n^{th}$ order Bessel Function of the First Kind and $a$ and $b$ are both positive real ...
5
votes
3answers
55 views
Integration question
I have trouble in integrating the following integral.
I would appreciate any help :D
$$\int_0^1 \sqrt{-\log x}\, a\, x^{a-1}dx$$
Thanks heaps :D
The answer is $\sqrt{\pi}/2(\sqrt{a})$.
4
votes
2answers
78 views
Integrating by substitution
I'm embarrassed to ask this question, but what's the flaw in the following evaluation?
$\displaystyle\int_{0}^{\pi} \sin (\sin x) \ dx = \int_{0}^{0} \frac{\sin u}{\sqrt{1-u^{2}}} \ du = 0$.
1
vote
2answers
163 views
Is the Riemann integral of a strictly positive function positive?
In the proof here a strictly positive function in $(0,\pi)$ is integrated over this interval and the integral is claimed as a positive number. It seems intuitively obvious as the area enclosed by a ...
3
votes
0answers
56 views
Simplify the integral with error function
$\newcommand{\erf}{\operatorname{erf}}$
I have the following integral and I need to simplify the solution. I have written first two steps. I don't know what is the value of
$$
\erf(\infty)
$$
I ...
0
votes
0answers
30 views
Solving an complex Integration with complex exp and other terms
I am trying to solve a partial differential equation and while solving I need to solve the following integral. If anyone could help me solve this integral that would be great.
$$y(x,t) = \int_{c-i ...
0
votes
2answers
40 views
Show that the integrals are equivalent
Show that:
$$\int_o^{\infty}\frac{\cos(x)}{1+x}dx=\int_o^{\infty}\frac{\sin(x)}{(1+x)^2}dx$$
I have no idea how to approach. The only thing I can think is substitution $y=\pi/2-x$ or integration by ...
1
vote
1answer
66 views
How to find limits of integration on a convolution of CRVs
In finding the convolution of two independent and continuous random variables, I am struggling with limits of integration. I cannot seem to figure out over what intervals the probability density ...
0
votes
0answers
30 views
Fundamental theorem of calculus 1 where integrand is a 2nd order partial derivative
I have a function $b(x,y)$ such that $b(x,0)=0$.
Now, suppose I wish to evaluate the following integral: (Note that $b$ is continuous almost everywhere but it is assumed that it is integrable. Also, ...
3
votes
1answer
114 views
Improper integral evaluation
I'm looking for a method to evaluate the following integral:
$\displaystyle \int_0^{\infty} \left( \frac{1}{e^x - 1} - \frac{1}{x} + \frac{e^{-x}}{2} \right) \frac{1}{x} dx$
EDIT:
Using the link, ...
2
votes
3answers
61 views
Definite Integral with a discontinuty
I have the next integral:
$$\int^{\pi/2}_0{\frac{\ln(\sin(x))}{\sqrt{x}}}dx$$
I have no clue how to start. At $x=0$ there is a clear discontinuity and I don't know how to solve the integral. The main ...
2
votes
2answers
56 views
Need to prove $\frac{3}{5}(2^{\frac{1}{3}}-1)\le\int_0^1\frac{x^4}{(1+x^6)^{\frac{2}{3}}}dx\le1$
I need to show that
$$\frac{3}{5}(2^{\frac{1}{3}}-1)\le\int_0^1\frac{x^4}{(1+x^6)^{\frac{2}{3}}}dx\le1$$
I just know that if in $[a,b]$, $f(x)\le g(x)\le h(x)$, then
...
1
vote
1answer
87 views
Evaluating the integral $\int_0^1\arctan(1-x+x^2)dx$
I need to evaluate
$$\int_0^1\arctan(1-x+x^2)dx$$
What I did: First I assume
$$I=\int_0^1\arctan(1-x+x^2)dx=\int_0^1\arctan((x-\frac{1}{2})^2+\frac{3}{4})dx$$
Since the function is symmetric about ...
0
votes
2answers
60 views
Definite integral of an exponential quotient
I was wondering if someone could help me find the definite integral of this:
$$ \int\limits_{R1}^{R2} \frac{t\, dt}{(t^2 + K^2)^{3/2}} $$
Where $\,K,\, R1,\, R2\,$ are constants, $\,R2>R1\,$ , ...
2
votes
1answer
126 views
A multiple integral question II
We know from the previous post that
$$\lim_{n\to\infty}\underbrace{\int_0^1 \int_0^1 \cdots \int_0^1}_{n \text{ times}}\frac{1}{(x_1\cdot x_2\cdots x_n)^2+1} ...
6
votes
1answer
90 views
A multiple integral question
Proving that
$$\lim_{n\to\infty}\underbrace{\int_0^1 \int_0^1 \cdots \int_0^1}_{n \text{ times}}\frac{1}{(x_1\cdot x_2\cdots x_n)^2+1} \mathrm{d}x_1\cdot\mathrm{d}x_2\cdots\mathrm{d}x_n=1$$
1
vote
2answers
54 views
Definite integral with functions in the sides
Im trying to resolve the next definite integral:
$$\int_{1-x^2}^{1+x^2}{\ln(t^2)\ dt}$$
Im not sure if I can use the Barrow's theorem, I think I have to use the fundamental theorem of integral ...
2
votes
0answers
89 views
Integral representation of Euler's constant
Prove that : $$ \gamma=-\int_0^{1}\ln \ln \left ( \frac{1}{x} \right) \ \mathrm{d}x.$$
where $\gamma$ is Euler's constant ($\gamma \approx 0.57721$).
This integral was mentioned in Wikipedia as in ...
9
votes
1answer
70 views
Prove that $\int_a^cf(x)\mathrm{d}x+(c-a)g(c)=\int_c^bg(x)\mathrm{d}x+(b-c)f(c)$
Let $f$ , $g$ be real continuous functions in $[a,b]$. Prove that there is $c\in(a,b)$ such that
$$\int_a^cf(x)\mathrm{d}x+(c-a)g(c)=\int_c^bg(x)\mathrm{d}x+(b-c)f(c)$$
What would you suggest me to ...
1
vote
3answers
89 views
Integration by parts question,, possibly a circular example [duplicate]
I am having trouble figuring this out.
$$\int_0^{1/3} \sec^3(\pi x) \, dx$$
We are currently doing integration by parts,, so I set $g(x)=\sec^3(\pi x)$ and $f'(x)=1$.
I arrived at: $$x\sec^3(\pi x) ...
8
votes
4answers
194 views
Calculate : $\int_1^{\infty} \frac{1}{x} -\sin^{-1} \frac{1}{x}\ \mathrm{d}x $
Find : $\displaystyle \int_1^{\infty} \frac{1}{x} -\sin^{-1} \frac{1}{x}\ \mathrm{d}x $.
I've done some work but I've got stuck, you may try to help me continue or give me another way , in both cases ...
2
votes
1answer
124 views
Characteristic function of the Smith-Volterra-Cantor set
Let the characteristic function of the SVC set be denoted by $ \beta $. Does the Riemann integral $ \displaystyle \int_{0}^{1} \beta ~ d{x} $ exist? I think it does since $ \beta $ is bounded, but I ...
17
votes
3answers
366 views
Evaluating the integral $\int_{-\infty}^\infty \frac {dx}{\cos x + \cosh x}$
Many recent questions have been asked here similar to this integral
$$\int_{-\infty}^\infty \frac {dx}{\cos x + \cosh x} = 2.39587\dots$$
whose "closed form" I cannot seem to figure out. I have ...
