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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2
votes
1answer
23 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
vote
1answer
36 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 ...
0
votes
1answer
20 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
vote
2answers
27 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
49 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
votes
1answer
26 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
vote
1answer
45 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?
0
votes
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
votes
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
vote
2answers
37 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
39 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 ...
1
vote
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?
0
votes
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
vote
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
vote
1answer
19 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
votes
1answer
38 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
vote
3answers
55 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
votes
1answer
27 views

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

Which type of functions will satisfy this? $$F: [1,\infty) \to [0,\infty)$$ $$\int_1^\infty \frac{F(y)}{y^2} dy \leq 1$$
1
vote
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
votes
1answer
14 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
votes
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
vote
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
votes
1answer
27 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 ...
0
votes
1answer
34 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
votes
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 ...
0
votes
2answers
19 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
votes
2answers
98 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
votes
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 $1$?
0
votes
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
vote
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
24 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
83 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 ...
1
vote
1answer
32 views

Polygon Matching [on hold]

Given a set of polygons vertices and a template polygon vertices, find all that match polygons from the given set of polygons with a template polygon.
2
votes
2answers
29 views

Evaluate 2D integral (by change of variable)

The question asks to evaluate integral $$\iint_D \Big[3-\frac12( \frac{x^2}{a^2}+\frac{y^2}{b^2})\Big] \, dx \, dy \ $$ where D is the region $$\frac{x^2}{a^2}+\frac{y^2}{b^2} \le 4 $$ I believe ...
-4
votes
0answers
18 views

Checking Fourier Series Problem [on hold]

I just worked out this sum. Please check this derivation.
1
vote
2answers
42 views

Prove the relation $\frac{1}{x}$=$\int^\infty_0$ $e^{-xt}$ dt, for $x>0$. Use it to prove $\int^\infty_0$ $\frac{\sin(x)}{x}$ dx = $\frac{\pi}{2}$

Prove the relation $$\frac{1}{x} = \int^\infty_0 e^{-xt}\, \text{d}t, \text{ for } x>0.$$ Use it to prove $$\int^\infty_0\frac{\sin(x)}{x}\, \text{d}x = \frac{\pi}{2}.$$ "Hint: Use ...
0
votes
3answers
27 views

Integral with polynomial

I have a problem with this this integral. $\int \frac x{x^2+2x+2} \,dx$ I know that result is connected with logarithm, because the numerator is derivative of denominator, but i can't figure how to ...
0
votes
1answer
39 views

Mean Value Theorem for Integrals

I understand their is an easy way of doing this, I just want to check if my working to a more complicated method is correct. $f$ continuous on $[a,b] \implies $ Riemann Integrable on $[a,b]$ ...
0
votes
0answers
23 views

Understanding Green's Theorem

When looking at Goursat's theorem in complex analysis, I came across the Wiki proof which involves beautiful application of Green's theorem. I saw Greens theorem simply as "connection between line ...
2
votes
0answers
29 views

Determine the volume of $A:=\{(x,y,z)\in \mathbb R^3 : \sqrt{x^2+y^2}\leq f(z)\}$

Let $f\in L^2(\mathbb R)$ and $f\geq0$. Determine $A:=\{(x,y,z)\in \mathbb R^3 : \sqrt{x^2+y^2}\leq f(z)\}$. "Normal" substitution $(x=rcos(\phi),y=rsin(\phi))$ did not help a lot, since I dont have ...
2
votes
2answers
58 views

Area enclosed by cardioid using Green's theorem

Let $$\gamma(t) = \begin{pmatrix} (1+\cos t)\cos t \\ (1+ \cos t) \sin t \end{pmatrix}, \qquad t \in [0,2\pi].$$ Find the area enclosed by $\gamma$ using Green's theorem. So the area enclosed by ...
0
votes
1answer
33 views

Trigonometric integral (arctg)

I have a problem with this integral. $$\int \text{arctan}(x-2)dx =\text{ }?$$ I tried integration by parts but it doesn't lead to right result.
0
votes
0answers
27 views

How do I verify that $\int_0^1 (1-t) \, f''(t) \, \mathrm dt = \int_x^{x+h} (x+h-u) \, f''(u) \, \mathrm du\;?$ [on hold]

How do I verify that: $$\int_0^1 (1-t) \, f''(ht+x) \, \mathrm dt = \int_x^{x+h} (x+h-u) \, f''(u) \, \mathrm du\;?$$
1
vote
4answers
24 views

How to caluclate the integral of $\int \frac{1}{\sqrt{4x^{2}+1}}dx$ using a trig substitution?

I am trying to determine the following integral: $\int \frac{1}{\sqrt{4x^{2}+1}} dx$ using a suitable substitution. My progress: let $x = \frac{1}{2} \tan \theta$ $dx = \frac{1}{2}\sec^{2} \theta ...