# Tagged Questions

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### Problem related to the integration

Recently I read that $$\int_{-p^{2}}^{+p^{2}} \frac{1}{\sqrt{x^{2}-p^{2}}}dx$$ tends to a finite real number as $p \to 0$. Can anyone explain me why this is true?
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### Show $\lim_{N\to\infty}\int_0^\pi\left(\frac1{\sin\frac{x}2}-\frac2x\right)\sin\left((N+\frac12)x\right)dx=0$

Prove that the function $\csc(x/2)-2/x$ is integrable on $(0,\pi)$. In fact, prove that it is bounded. In fact, prove that it tends to zero as $x\to0$. Use this to show that ...
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$\displaystyle \int\frac{dx}{(x^2-x+1).\sqrt{x^2+x+1}}$ My solution :: $\displaystyle \int\frac{1}{(x^2-x+1)^{\frac{3}{2}}.\sqrt{\frac{x^2+x+1}{x^2-x+1}}}dx$ Now Let $\displaystyle ... 1answer 203 views ### Determine if the following integral is convergent or divergent. If it is convergent find its value. [duplicate] The next week I have a test over Improper Integrals and yesterday started to solve some problems that I've found on th Internet. Everything was fine until I stuck on this example: Determine if the ... 1answer 89 views ### Integral of a series [closed] I cannot solve this integral, can anyone help me? $$\int_0^\infty \left(x^3 \sum_{n=1}^{+ \infty} e^{-nx} \right)dx$$ Thank you in advance 2answers 225 views ### How to prove that$\int_0^{2\pi} \sqrt{1+\cos^2{t}}\; dt>2\pi$? Let $$I=\int_0^{2\pi} \sqrt{1+\cos^2{(t)}}\;dt$$ How to prove, in an elementary way, that$I>2\pi$? 0answers 342 views ### Integral with Legendre polynomials What is $$\int_{\frac{\pi}{2}}^{\pi} P_l(\cos(\theta))P_{l'}(\cos(\theta)) \sin(\theta) d\theta$$ in general? Of course,$P_l$is the l-th Legendre polynomial. Obvisiouly it is for even/even and ... 3answers 95 views ### Can we use the Second Mean Value Theorem over infinite intervals? Let$[a,b]$be any closed interval and let$f,g$be continuous on$[a,b]$with$g(x)\geq 0$for all$x\in[a,b]$. Then the Second Mean Value Theorem says that $$\int_a^bf(t)g(t)\text{d}t = ... 3answers 230 views ### the value of \lim\limits_{n\rightarrow\infty}n^2\left(\int_0^1\left(1+x^n\right)^\frac{1}{n} \, dx-1\right) This is exercise from my lecturer, for IMC preparation. I haven't found any idea. Find the value of$$\lim_{n\rightarrow\infty}n^2\left(\int_0^1 \left(1+x^n\right)^\frac{1}{n} \, dx-1\right)$$... 1answer 210 views ### How do I integrate \int_{0}^{1}\!\sin x^2\,dx? How do I integrate$$ \int_{0}^{1}\!\sin x^2\,dx? $$Will it be so complicated? 2answers 106 views ### Show that \int_{(0,1)\times (0,1)} \frac{1}{1-xy} dxdy = \sum_{n=1}^{\infty} \frac{1}{n^2} I'm looking for a clever way to show that$$ \int\limits_{(0,1)\times (0,1)} \frac{1}{1-xy} dxdy = \sum_{n=1}^{\infty} \frac{1}{n^2}.$$All suggestions will be appreciated! 1answer 210 views ### Qualifying problem for real analysis: limit involving definite integral The following problem has appeared in 2013 January qualifying exam in Purdue University, which is publicly available here. Problem 3. Let \{a_k\} be sequence of positive numbers such that ... 1answer 261 views ### A challenging logarithmic integral \int_0^1 \frac{\log(1+x)\log(1-x)}{1+x}dx While playing around with Mathematica, I found that$$\int_0^1 \frac{\log(1+x)\log(1-x)}{1+x}dx = \frac{1}{3}\log^3(2)-\frac{\pi^2}{12}\log(2)+\frac{\zeta(3)}{8}$$Please help me prove this result. 1answer 63 views ### Test of convergence of \int_{-\infty}^{\infty} \dfrac{x^6+6}{x^8+8}dx I am having some trouble with this problem and don't know if I am doing it right:$$\int_{-\infty}^{\infty} \dfrac{x^6+6}{x^8+8}dx$$so the steps I have taken so far are, I split it into ... 2answers 49 views ### How to calculate \lim_{x\to 1}\int_{0}^{1}\frac{dy}{\sqrt{1-y^{2}}}\frac{y^{3/2}}{\sqrt{x - y}} when x>1? Numerically, it looks that the limit is$$\lim_{x\to 1}\int_{0}^{1}\frac{dy}{\sqrt{1-y^{2}}}\frac{y^{3/2}}{\sqrt{x - y}} = \frac{1}{\sqrt{2}}\log(1 - x) + cte $$, but I have not been able to ... 2answers 241 views ### Trig Fresnel Integral$$\int_{0}^{\infty }\sin(x^{2})dx$$I'm confused with this integral because the square is on the x, not the whole function. How can I integrate it? Thank you. I have not done complex analysis (only ... 1answer 91 views ### integration as limit of a sum If f is continuous on [0, 1] then$$\lim_ {n\to\infty}\sum_{j=0}^{\lfloor n/2\rfloor} \frac1{n}f\left(\frac {j}{n}\right) = ? $$will the answer be that the limit exists and is equal to ... 1answer 110 views ### Proof that Riemann-integrability is preserved by changing the function at a converging sequence of points Let (x_n)^{\infty}_{n=1}\in[a,b] such that \lim\limits_{n\to\infty}x_n=l. Let g:[a,b]\to \mathbb{R} be bounded function. Assume that there exists a Riemann integrable function f:[a,b]\to ... 1answer 178 views ### How to prove \frac{\pi^2}{6}\le \int_0^{\infty} \sin(x^{\log x}) \ \mathrm dx ? I want to prove the inequality$$\frac{\pi^2}{6}\le \int_0^{\infty} \sin(x^{\log x}) \ \mathrm dx $$There are some obstacles I face: the indefinite integral cannot be expressed in terms of ... 0answers 146 views ### Riemann sums vs Darboux sums Let we speak of the tagged partitions of an interval and a bounded function defined on it. I think that the tags give rise to particular Riemann sums which may be quite different in value that the ... 0answers 31 views ### An integral inequality related to Taylor expansion Problem. Let f:[a,b]\to\mathbb{R} be a function such that f\in C^3([a,b]) and f(a)=f(b). Prove that$$ ... 2answers 85 views ### Asymptotic expansion for$\frac{1}{2\zeta(3)}\int_x^\infty \frac{u^2}{e^u - 1} du$? Is there an asymptotic expansion for the function: $$g(x)=\frac{1}{2\zeta(3)}\int_x^\infty \frac{u^2}{e^u - 1} du,$$ over the domain$x\in [0,\infty)$in terms of ... 1answer 61 views ### Further explanation needed for “If$f_n(x) \longrightarrow 0$almost everywhere (a.e.) and…” I came across the following problem and do not know how to proceed: Let$\{f_n\}$be a sequence of integrable functions defined on an interval$[a,b]$. Then I have to prove that "If ... 2answers 268 views ### How to calculate$\int_{0}^{1}(\arcsin{x})(\sin{\frac{\pi}{2}x})dx$? How to find the follwing integral's value ? $$\int_{0}^{1}(\arcsin{x})(\sin{\frac{\pi}{2}x})dx$$ Actually, I don't know it can be represented as closed form. 1answer 64 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, ... 3answers 496 views ### The equivalence between Cauchy integral and Riemann integral for bounded functions Definitions Suppose$P\colon a=x_0<x_1<\dotsb<x_n=b$is a partition of$[a,b]$. Let$\Delta x_k=x_k-x_{k-1}$and$\lVert P\rVert$denotes$\max_{0<k\le n}\Delta x_k$. The Cauchy integral ... 6answers 518 views ### Alternative solutions to$\lim_{n\to\infty} \frac{1}{\sqrt{n}}\int_{ 1/{\sqrt{n}}}^{1}\frac{\ln(1+x)}{x^3}\mathrm{d}x$Here is a limit that can be computed directly by performing the integration and then taking the limit, but the way is rather ugly. What else can we do? Might we avoid the integration? ... 1answer 34 views ### A small clarification Could you please explain the following result? $$n\left(\frac{1}{n}\sum_{i=1}^nf \left(\frac{i}{n}\right)-\sum_{i=1}^n \int^\frac{i}{n}_\frac{i-1}{n}f(x)dx ... 2answers 153 views ### Riemann Integral Q. Compute the following limit:$$\lim_{n\rightarrow\infty}\left(\frac{2^{1/n}}{n+1}+\frac{2^{2/n}}{n+1/2}+\cdots+\frac{2^{n/n}}{n+1/n}\right)$$using integral. One obvious method is to squeeze it ... 1answer 156 views ### A multiple integral question II We know from the previous post that$$\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 104 views ### A multiple integral question Proving that $$\lim_{n\to\infty}\underbrace{\int_0^1 \int_0^1 \cdots \int_0^1}_{n \text{ times}}\frac{1}{(x_1\cdot x_2\cdots x_n)^2+1} \mathrm{d}x_1\cdot\mathrm{d}x_2\cdots\mathrm{d}x_n=1$$ 1answer 144 views ### limit of the error in approximating definite integral with midpoint rule I want to calculate$\lim_{n \rightarrow \infty} n^2 |\int_{[0,1]}f(x)-I_n(x)|$where$I_n$is the integral approximation by midpoint rule:$I_n=\frac{1}{n}\sum_{k=1}^nf(c_k)$and$c_k$is the point ... 2answers 172 views ### Evaluation of$\lim_{n\to\infty} \int_0^1 \frac{e^{\displaystyle x^{n}}}{1+x^2}\,\mathrm{d}x$Evaluation of $$\lim_{n\to\infty} \int_0^1 \frac{e^{\displaystyle x^{n}}}{1+x^2}\,\mathrm{d}x$$ Sis. 2answers 606 views ### Evaluation of$\int_0^{\pi/3} \ln^2\left(\frac{\sin x }{\sin (x+\pi/3)}\right)\,\mathrm{d}x$I plan to evaluate $$\int_0^{\pi/3} \ln^2\left(\frac{\sin x }{\sin (x+\pi/3)}\right)\, \mathrm{d}x$$ and I need a starting point for both real and complex methods. Thanks ! Sis. 2answers 143 views ### How to evaluate a definite integral that involves$(dx)^2\$?
For example: $$\int_0^1(15-x)^2(\text{d}x)^2$$