# Tagged Questions

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 ?
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
58 views

84 views

### Is it possible to evaluate this integral? [duplicate]

Is it possible to evaluate this integral: $$\int_{0}^{\frac{\pi }{2}}\ln(\sin 2x){\rm d}x$$
$$\int_{0}^{\frac{\pi }{2}}x\cot(x)dx$$ 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 ...
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 $\quad ... 3answers 35 views ### Confusing Triple Integral i'm having trouble with this integral the integral is$\int_0^9\int_{\sqrt z}^3\int_0^y z\cos(y^6)\,dx\,dy\,dz$. We aren't given any more information and i'm a bit stuck as to where to start. I don't ... 2answers 53 views ### Triple integral problem involving a sphere Let$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 ... 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 ... 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 ... 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$1answer 25 views ### definite integral negative variable Man, it's been so long since I did this. I am trying to do this: NB: limits are$-\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$... 3answers 86 views ### Is this definite integral impossible? From my understanding when you integrate$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 ... 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: $$|a|<1$$ $$x\in [-\pi,\pi]$$ $$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$ ... 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: $$|a|<1$$ $$x\in [-\pi,\pi]$$ $$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$ ... 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 ... 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 ... 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 ... 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 ... 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. $$\displaystyle ... 2answers 161 views ### Let f:[a,b]\to\mathbb R be Riemann integrable and f>0. Prove that \int_a^bf>0. (No Measure theory) [closed] Is the Riemann integral of a strictly positive function positive? This is not a duplicate. I'm specifically interested in a proof not involving Measure Theory. The thread above uses the fact that f ... 0answers 105 views ### Riemann sums vs Darboux sums Let we speak of the tagged partitions of an interval and a bounded function defined on it. I think that the tags give rise to particular Riemann sums which may be quite different in value that the ... 3answers 77 views ### Integral with Bessel functions of the First Kind. I'd like to solve the following integral: I = \int_0^\infty J_0(at) J_1(bt) e^{-t} dt\ where J_n is an n^{th} order Bessel Function of the First Kind and a and b are both positive real ... 3answers 55 views ### Integration question I have trouble in integrating the following integral. I would appreciate any help :D$$\int_0^1 \sqrt{-\log x}\, a\, x^{a-1}dx$$Thanks heaps :D The answer is \sqrt{\pi}/2(\sqrt{a}). 2answers 78 views ### Integrating by substitution I'm embarrassed to ask this question, but what's the flaw in the following evaluation? \displaystyle\int_{0}^{\pi} \sin (\sin x) \ dx = \int_{0}^{0} \frac{\sin u}{\sqrt{1-u^{2}}} \ du = 0. 2answers 163 views ### Is the Riemann integral of a strictly positive function positive? In the proof here a strictly positive function in (0,\pi) is integrated over this interval and the integral is claimed as a positive number. It seems intuitively obvious as the area enclosed by a ... 0answers 56 views ### Simplify the integral with error function \newcommand{\erf}{\operatorname{erf}} I have the following integral and I need to simplify the solution. I have written first two steps. I don't know what is the value of$$ \erf(\infty) $$I ... 0answers 30 views ### Solving an complex Integration with complex exp and other terms I am trying to solve a partial differential equation and while solving I need to solve the following integral. If anyone could help me solve this integral that would be great.$$y(x,t) = \int_{c-i ... 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 ... 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 ... 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, ... 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, ... 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 ... 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 ... 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 ... 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,$\,R2>R1\,$, ... 1answer 126 views ### A multiple integral question II We know from the previous post that $$\lim_{n\to\infty}\underbrace{\int_0^1 \int_0^1 \cdots \int_0^1}_{n \text{ times}}\frac{1}{(x_1\cdot x_2\cdots x_n)^2+1} ... 1answer 90 views ### A multiple integral question Proving that$$\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$$2answers 54 views ### Definite integral with functions in the sides Im trying to resolve the next definite integral:$$\int_{1-x^2}^{1+x^2}{\ln(t^2)\ dt}$$Im not sure if I can use the Barrow's theorem, I think I have to use the fundamental theorem of integral ... 0answers 89 views ### Integral representation of Euler's constant Prove that :$$ \gamma=-\int_0^{1}\ln \ln \left ( \frac{1}{x} \right) \ \mathrm{d}x.$$where \gamma is Euler's constant (\gamma \approx 0.57721). This integral was mentioned in Wikipedia as in ... 1answer 70 views ### Prove that \int_a^cf(x)\mathrm{d}x+(c-a)g(c)=\int_c^bg(x)\mathrm{d}x+(b-c)f(c) Let f , g be real continuous functions in [a,b]. Prove that there is c\in(a,b) such that$$\int_a^cf(x)\mathrm{d}x+(c-a)g(c)=\int_c^bg(x)\mathrm{d}x+(b-c)f(c)$$What would you suggest me to ... 3answers 89 views ### Integration by parts question,, possibly a circular example [duplicate] I am having trouble figuring this out.$$\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) ... 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 $. I've done some work but I've got stuck, you may try to help me continue or give me another way , in both cases ... 1answer 124 views ### Characteristic function of the Smith-Volterra-Cantor set Let the characteristic function of the SVC set be denoted by$ \beta $. Does the Riemann integral$ \displaystyle \int_{0}^{1} \beta ~ d{x} $exist? I think it does since$ \beta $is bounded, but I ... 3answers 366 views ### Evaluating the integral$\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 ...