Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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2answers
24 views

Complicated integration

How can this be integrated? : $$\int_{b}^{a} x \left ( \frac{D}{a-b} \right ) \left ( \frac{a-x}{a-b} \right )^{D-1}dx$$ The solution is : $$\frac{a+(D)(b)}{D+1}$$
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1answer
19 views

Integration of exponential function (problem below):

I have tried u substitution, but I'm not really sure what to do. This is what I tried: $u = x^3 - 2x^2 + x + 7\\ du = (3x^2-4x+1)dx$ But I'm not sure what to do now. I also tried: $u = 2x-1\\ ...
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0answers
17 views

Whats the most we can say about two functions with the same integral?

Okay so if $f'(x) = g'(x)$ then $f(x) = g(x) + C$ and I understand that graphically but if $\int f(x)dx = \int g(x)dx$ then it is not necessary the case that $f(x) = g(x) + C$ even if both the ...
8
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2answers
42 views

How do I evaluate the integral $\int \frac{1}{x^3 -1}dx$

this is my first question here so I hope I did everything right. Still really new to LaTeX as well$$\int \frac{1}{x^3 -1}dx $$ I first used partial fractions to decompose this integral into two ...
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0answers
23 views

$\int_0^x \frac{\cos(t)}{\sin(t)+\cos(t)} dt$ and $\int_0^x \frac{\sin(t)}{\sin(t)+\cos(t)} dt$

Let $x\in[0,\pi/2]$, $I(x)=\int_0^x \frac{\cos(t)}{\sin(t)+\cos(t)} dt$ and $J(x)=\int_0^x \frac{\sin(t)}{\sin(t)+\cos(t)} dt$ . The aim of the exercise is to calculate $I(x)$ and $J(x)$. Fisrt of ...
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1answer
23 views

Proving that the integral of $\cos^m(x)\sin{(nx)}$ between $0$ and $\pi$ is zero

I've been doing a question that initially asks to derive a reduction formula for the indefinite integral of $\cos^m(x)\sin{(nx)}$ then the next part asks to prove that: ...
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3answers
37 views

Prove that $f(x)$ is differentiable at any point

Given this condition: $x^2 \sin(x^2) \le x^3f(x)\le \sin(x^4)$ at any open interval that goes through $0$ I need to prove that $f(x)$ is differentiable at $x=0$, but I couldn't come up how, it ...
1
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0answers
15 views

Expected value of an expected value of a joint density function

I had a question I was hoping for some help on: Let $Y_1$ and $Y_2$ be continuous random variables with joint density function: $$f(x,y) = \begin{cases} 6(1-y_2) & \text{if $0 <= y_1 <= ...
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1answer
40 views

Problem on Integration: $\Bbb R-\Bbb C$ split and pull back of forms

This post is not short. However I'm sure that a guy who good handle these concepts, could read and answer in five minutes. I only want to write my attempt, in order to understand where I'm wrong. Let ...
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4answers
71 views

A level Integration

Using the substitution $x=\cosh (t)$ or otherwise, find $$\int\frac{x^3}{\sqrt{x^2-1}}dx$$ The correct answer is apparently $$\frac{1}{3}\sqrt{x^2-1}(x^2+2)$$ I seem to have gone very wrong ...
1
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1answer
33 views

Surface/Path Integral Approach - Brain Fart?

Many times I have dealt with path and surface integrals of the following form $$\int_C \mathbf{F}\cdot d\mathbf{r} \,\,\,\,\,\textrm{(path integral)}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\int_S ...
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1answer
28 views

How to compute $\int_{0}^{\infty}dx\:\frac{\exp(-ax^2+bx)}{x+1}\:\text{ for }\: a>0, b\in \mathbb{C}$?

As the title says I am trying to compute the integral $I=\displaystyle\int_{0}^{\infty}dx\:\frac{\exp(-ax^2+bx)}{x+1}$ where $a>0$ and $b$ is a complex number. For the special case of $b=-2a$, we ...
3
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3answers
57 views

Convergence testing of the improper integral $\int_{0}^{\infty}\frac{\ln x}{\sqrt{x}(x^2-1)}\ dx$

I've tried to test this integral for convergence for a couple of hours, actually I know that $$\int_{2}^{\infty}\frac{\ln x}{\sqrt{x}(x^2-1)}\ dx$$ converges with no problem with the help of Dirichlet ...
1
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1answer
42 views

How to show $\int_0^{\infty} \frac{1}{\sqrt{x}}\sin({\frac{1}{x}})dx$ converges

How to show $\int_0^{\infty} \frac{1}{\sqrt{x}}\sin({\frac{1}{x}})dx$ converges? I have that $$\frac{-1}{\sqrt{x}}\le \frac{\sin({\frac{1}{x}})}{\sqrt{x}} \le \frac{1}{\sqrt{x}}$$ but when you ...
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1answer
23 views

General question about simplification

After done with integration I got the final answer as: $\ln(a+4) + \ln(a-4) + C$ I can rewrite it as: $\ln((a+4)(a-3)) + C$ But in book it is written as: $\ln((a+4)(a-3)+C)$ Is it correct? and ...
1
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2answers
35 views

Surface Area by Integration

$$2\pi\int_{3}^6\left(\frac{1}{3}x^\frac{3}{2}-x^\frac{1}{2}\right)\left(1+\left(\frac{1}{2}x^\frac{1}{2}-\frac{1}{2}x^\frac{-1}{2}\right)^2\right)^\frac{1}{2}dx$$ I've managed to simplify this down ...
3
votes
2answers
50 views

Find $\lim \limits_{x \to \pi}\frac{\int_0^x\cos^2(t)dt}{x-\pi}\;$

$$\lim \limits_{x \to \pi}\frac{\int_0^x\cos^2(t)\,dt}{x-\pi}$$ I don't understand why the limit is not $\infty$ How is the limit: $1$?
3
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1answer
27 views

Convergence of series of integrals

Let $\phi \in C^\infty(\mathbb R)$ be a function such that $\phi(x), \phi'(x) \to 0$ as $x \to \infty$. I want to show that $$\lim_{n \to \infty} \int_\mathbb R \cos(nx) \phi(x) \ dx = 0$$ Doing it ...
1
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1answer
47 views

Find $\lim \limits_{x \to 0} \frac{\int_0^x \frac{t\,dt}{\cos t}}{\sin^2(x)}\,$

$$\lim_{x \to 0} \dfrac{\int_0^x \frac{t\,dt}{\cos t}}{\sin^2(x)}$$ what does it mean when the limit of $x$ is $0$ in the integral? How do I calculate this limit?
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0answers
51 views

Integration over a variable

Can someone explain to me the step by step of this integration? $$∫_0^r(a-\frac{r}{b})dv$$ Where $v$ is the volume of a cylinder $ \pi r^2h$ The answer is $$ \frac{a}{2}-\frac{r}{3b}$$ But it's ...
3
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2answers
48 views

Evaluating the Definite Integral $\int_0^{\pi}\cos^{2n} \theta d\theta$

$$\int_0^{\pi}\cos^{2n} \theta d\theta$$ $$u=\cos \theta \implies du= -\sin \theta d\theta \implies d\theta= -\frac{du}{1-u^2} $$ $$\int_{-1}^1 \frac{u^n}{1-u^2} du=\int_{-1}^1 ...
1
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2answers
38 views

Finding the value of $3(\alpha-\beta)^2$ if $\int_0^2 f(x)dx=f(\alpha) +f(\beta)$ for all $f$

Let $f$ be a polynomial of degree $n$ at most $3$ such that there exists some $\alpha,\beta$ satisfying $\int_0^2 f(x)dx=f(\alpha) +f(\beta)$ for all such $f$. Find the value of $3(\alpha-\beta)^2$ ...
3
votes
1answer
41 views

Need help with continuing an idea concerning showing that $4\sum\limits_{n \ge 1} a_n^2 \ge \sum\limits_{n \ge 1} \frac1{n^2}(a_1+…+a_n)^2 $

I recently encountered the following problem: If $\sum a_n^2 $ converges and $\alpha_n= \frac{a_1+...+a_n}{n}$ then show that: $$4\sum_{n \ge 1} a_n^2 \ge \sum_{n \ge 1} \alpha_n^2$$ I had an ...
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2answers
29 views

Find $\int (e^{2x}+e^{3x})^\frac{1}{2}dx$

$$\int (e^{2x}+e^{3x})^\frac{1}{2}dx$$ I'm not sure what substitution I'm supposed to make here. Can someone help?
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0answers
19 views

Solving System of 2 simple odes

I am just trying to solve two simple odes using Runge-Kutta method: \begin{equation} \frac{dx}{dt} = v \end{equation} \begin{equation} m .\frac{dv}{dt}= f_{1}(x)+f_{2}(x,v) \end{equation} ...
1
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1answer
30 views

How to find $\int_0^{1/4}\frac{1}{x\sqrt{1-4x}}\ln\left({\frac{1+\sqrt{1-4x}}{2\sqrt{1-4x}}}\right)dx$

Let $H_n$ be the harmonic series. I want to find the value of $A=\displaystyle\sum_{n=0}^\infty \binom{2n}{n}\left(\frac{1}{4}\right)^n\frac{H_n}{n} $. From this paper : ...
1
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1answer
20 views

Are these two elliptic integral evaluations identical?

I'm reading a paper on the Schwarz D minimal surface, and I'm wondering whether the authors have made a mistake. They evaluate the integral $$ \int_0^z \frac{2t\;\mathrm{d}t}{\sqrt{t^8-14t^2+1}}, $$ ...
2
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1answer
46 views

Find $\int_0^1 \frac{dx}{(1+x^n)^2\sqrt[n]{1+x^n}}$

Find $$\int_0^1 \frac{dx}{(1+x^n)^2\sqrt[n]{1+x^n}}$$ with $n \in \mathbb{N}$. My tried: I think that, needing to find the value of $$I_1=\int_{0}^1 \frac{dx}{(1+x^n)\sqrt[n]{1+x^n}}$$ because: ...
1
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3answers
56 views

$\int^\infty_0 \frac{\cos(x)}{\sqrt{x}}\,dx$ Evaluate using Fresnel Integrals

$\int^\infty_0 \frac{\cos(x)}{\sqrt{x}}\,dx$ Evaluate using Fresnel Integrals (For reference the $\cos$ Fresnel integral is $\int^\infty_0 \cos(x^2)\, dx = \frac{\sqrt{2 \pi}}{4}$) I've tried ...
0
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1answer
27 views

What does it mean $\int_1^\infty\frac{F(y)}{y^2}\mathrm dy$?

Which type of functions will satisfy this? $$F: [1,\infty) \to [0,\infty)$$ $$\int_1^\infty \frac{F(y)}{y^2} dy \leq 1$$
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1answer
57 views

Solving integral $\int \arcsin x \cos x dx$ [on hold]

Can anyone give me a hint how to solve $\int \arcsin(x)\cos(x)dx $ ?
0
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1answer
15 views

How to plot function of three or more variable?

How to plot function of two or more variable ? Also,why do we require perpendicular axis for the function to be examined ?
3
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2answers
49 views

How can I show that $f$ must be zero if $\int fg$ is always zero?

Let $f(x)$ be continuous on $[a,b]$ and suppose $\int_a^b f(x)g(x)dx = 0$ for every continuous function $g$ on $[a,b]$. Prove that $f(x)=0$ on $[a,b]$. I understand that $f(x)$ must be zero ...
1
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0answers
22 views

Proving that $\int_{\mathbb{R}} f \ d\mu = \frac{1}{N}\sum_{i=1}^N f(\lambda_i)$

I want to know if my proof is correct and if there is some easier way to prove this (you don't need to read all my proof, I'm accepting as answers another proofs, not just corrections of mine). ...
0
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1answer
28 views

Integration Convergence/Divergence Questtion

$$ \int\limits_0^{\pi} \frac{ dt}{\sqrt{t} + \sin t }$$ How can one tell if this integral converges or diverges? Integral of $1/(\sqrt{t}+\sin(t))$ from $0$ to $\pi$. I can't even find the ...
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1answer
35 views

Prove that $\int_0^{\pi} \sin^nx\sin(n+2)xdx=\int_{0}^{\pi}\sin^nx\cos(n+2)xdx=0$

Prove that $$\int_0^{\pi} \sin^nx\cdot\sin(n+2)xdx=\int_{0}^{\pi}\sin^nx\cdot\cos(n+2)xdx=0$$ with $n \in \mathbb{N}$ I think it's true, but I can't prove.
2
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1answer
18 views

Integration by parts $\int(x+y)e^{-x}dx$

What I'm trying to solve: $\int(x+y)e^{-x}dx$ Here's my professor's approach: $$u = x, du = e^{-x}$$ $$du = dx, dv = -e^{-x}$$ By doing parts: $(-xe^{-x}) - \int(-e^{-x})dx - ye^{-x} = (-xe^{-x}) ...
3
votes
1answer
42 views

How to use U substitution for the integral $\int\frac{8}{49+x^2}\,dx$?

So, the following is my problem. $$\int\frac{8}{49+x^2}\,dx$$ I understand this. I should first take out the constant which is 8 so it'll be $$8\int\frac{1}{49+x^2}\,dx$$ Then I should factor out the ...
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2answers
20 views

Integration inequality question help: Sketch the curve y=1/u for u > 0…

Sketch the curve $y=\frac{1}{u}$ for $u > 0$. From the diagram, show that $\int\limits_1^{\sqrt{x}}\frac{du}{u}< \sqrt x-1$, for x > 1. Use this result to show that $0 < \ln(x) < ...
4
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2answers
103 views

Integration help - question: $e^{-\sin(x)}$

I would really like some help with the integration of $e^{-\sin(x)}$. Thanks to anyone who will help :) Given that $\sin(x) > \frac{2x}{\pi}$ for $0 < x < \frac{\pi}{2}$, where ...
3
votes
1answer
28 views

Why does $\int_b^{b+\Delta b}f(x)\;dx=f(b)\Delta b+\mathcal{O}(\Delta b^2)$

On this page it is shown that: $$\frac{\partial}{\partial b}\left(\int_a^bf(x)\;dx\right)=\lim_{\Delta b\rightarrow 0}\frac{1}{\Delta b}\int_b^{b+\Delta b}f(x)\;dx=\lim_{\Delta b\rightarrow ...
3
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1answer
48 views

Why does the integral equal $1$?

Let $a\in\mathbb{R}-\mathbb{Z}$. Why is the following equality true? $$1 = \frac{1}{2\pi} \int_0^{2\pi} \left| e^{-i(\pi-x)a} \right|^2 dx$$ More precisely, why is the integrand equals $1$?
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1answer
23 views

help with improper integral and cuberoot? [on hold]

Evaluate the following integral . $$\int_{-\infty}^{0} \frac{1}{(x+2)^{1/3}}dx$$
1
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0answers
20 views

FP3 Integration help [duplicate]

$$I_{n}=\int x^n(1-x^2)^{\frac{1}{2}} dx$$ Show that $$(n+2)I_{n}=(n-1)I_{n-2}-x^{n-1}(1-x^2)^{\frac{3}{2}}$$ So far I have done this: $$\int x^{n-1}(x)(1-x^2)^{\frac{1}{2}} dx$$ $$u=x^{n-1}$$ ...
0
votes
1answer
36 views

Reduction formulae

$$I_{n}=\int x^n(1-x^2)^{\frac{1}{2}} dx$$ Show that $$(n+2)I_{n}=(n-1)I_{n-2}-x^{n-1}(1-x^2)^{\frac{3}{2}}$$ I can't seem to get this answer. Can someone please explain how to get to this? Thanks ...
2
votes
2answers
83 views

Solving an integral (with substitution?)

For a physical problem I have to solve $\sqrt{\frac{m}{2E}}\int_0^{2\pi /a}\frac{1}{(1-\frac{U}{E} \tan^2(ax))^{1/2}}dx $ I already tried substituting $1-\frac{U}{E}\tan^2(ax)$ and ...
3
votes
0answers
25 views

Understanding averaging of symplectic matrices via Haar measure

In McDuff and Salamon's Intro. to Symplectic Topology (2nd edition), there's a proof that $U(n)$ is a maximal compact subgroup of $Sp(2n)$ which I'm trying to understand. The proof uses the Haar ...
0
votes
1answer
20 views

Integral evaluation involving trignometric functions

How to explain the following equality? (Part of an integral calculation): $$\frac{2}{2\pi}\int_{-\pi}^\pi \left| \sin x \right| (\cos nx + i\sin nx) dx = \frac{4}{2\pi}\int_0^{2\pi} \sin x \cos nx ...
0
votes
0answers
29 views

Proof of the Poincare inequality for $W_0^{1,2}((a,b))$.

I have a question about an exercise for which I already have the solution, which I do not unterstand completely. Let $a, b \in \mathbb R$ with $0 < a < b$. Then we have \begin{align*} ...
4
votes
3answers
86 views

Evaluation of integral $\int_{0}^{\infty}\frac{\sin x}{x\left ( 1+x^2 \right )^2}\,{\rm d}x$

I'm trying to evaluate the following integral: $$\mathcal{J}=\int_{0}^{\infty}\frac{\sin x}{x\left ( 1+x^2 \right )^2}\,{\rm d}x$$ Well there are $3$ poles , one lying on the real line the other on ...