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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8
votes
2answers
160 views

Closed form of $\int_0^1\int_0^1\int_0^1\frac{\left(1-x^y\right)\left(1-x^z\right)\ln x}{(1-x)^3}\,\mathrm dx\;\mathrm dy\;\mathrm dz$

While trying to find several references to answer Pranav's problem, I encounter the following multiple integrals $$I=\int_0^1\int_0^1\int_0^1\frac{\left(1-x^y\right)\left(1-x^z\right)\ln ...
0
votes
0answers
10 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*} ...
1
vote
2answers
30 views

^2Evaluation 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 ...
2
votes
2answers
21 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 ...
0
votes
0answers
9 views

Polygon Matching

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.
-4
votes
0answers
12 views

Checking Fourier Series Problem

I just worked out this sum. Please check this derivation.
0
votes
0answers
48 views

Find the volume bounded by the sphere $x^2+y^2+z^2=4$ and $x^2+y^2-2x=0$

This question appeared on my calculus exam yesterday. I don't know how to do it: Find the volume bounded by the sphere $x^2+y^2+z^2=4$ and $x^2+y^2-2x=0$. My attempt: First, I realised that the ...
1
vote
2answers
37 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
4answers
24 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 ...
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
28 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 $[a,b]$ ...
11
votes
3answers
259 views

How to prove $\int_0^1\frac{x^3\arctan x}{(3-x^2)^2}\frac{\mathrm dx}{\sqrt{1-x^2}}=\frac{\pi\sqrt{2}}{192}\left(18-\pi-6\sqrt{3}\,\right)$?

How to prove the following result? $$\int_0^1\frac{x^3\arctan x}{(3-x^2)^2}\frac{\mathrm dx}{\sqrt{1-x^2}}=\frac{\pi\sqrt{2}}{192}\left(18-\pi-6\sqrt{3}\,\right)$$ For my part no idea?
4
votes
2answers
75 views

Why does $\int_0^1 \frac 1 { \sqrt{ x (1 - x) } } \, \mathrm d x = \pi$?

I was wondering why the following is true: $$\int_0^1 \frac 1 { \sqrt{ x (1 - x) } } \, \mathrm d x = \pi$$ It is easy to obtain this result by doing a trig substitution but it's messy and not ...
2
votes
0answers
28 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 ...
0
votes
0answers
19 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 ...
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
26 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
23 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 ...
3
votes
0answers
21 views

Issues proving a basis via wedge product

On a quiz I was given the problem" a series that is a basis for $[-1,1]$ is $ \sum_0^{\infty} c_n P_n $, where $ P_n $ is a polynomial and each polynomial $P_n$ is orthonormal to the others. Using the ...
2
votes
2answers
59 views

How to solve such an integration analytically?

$\displaystyle\int^{2\pi}_{0} e^{ia \cos{\theta}}d\theta$ where $a$ is some constant. Can it be solved with some substitution? I tried it by expanding the exponential series but that was not proper ...
2
votes
1answer
28 views

Evaluate $\int^\infty_0 t^{a+b-1}(t+1)^{-b-1} U(a+2,a-b+2,ct)dt$

Evaluate $$ \int^\infty_0 t^{a+b-1}\left(t+1\right)^{-b-1} U\left(a+2,a-b+2,ct\right)dt $$ under the condition $a>0$, $b>0$ and $c>0$, where $U(\cdot,\cdot,\cdot)$ denotes the ...
0
votes
1answer
20 views

Volumes of revolution?

The point $P(a,b)$ lies on the curve $y=\mathrm{arsinh}\,x$. $R$ is the region bounded by the curve, the $x$- and $y$-axes and the line $x=a$. When $R$ is rotated $2\pi$ radians about the $x$-axes the ...
0
votes
0answers
18 views

Volumes of revolutions question

The point $P(a,b)$ lies on the curve $y=arsinh x$. $R$ is the region bounded by the curve, the $x$- and $y$-axes and the line $x=a$. When $R$ is rotated 2$π$ radians about the $x$-axes the solid ...
0
votes
0answers
23 views

Change of variable of integration, for numerical integration

I have an independent (array) variable $r = {r_0, r_1, ..., r_N}$, and three functions (arrays) of that variables, $n(r) ={n_0, n_1, ..., n_N}$, $p(r)$, and $E(r)$. How can I calculate the function ...
0
votes
1answer
21 views

finding the parametric path for line integral

Calculate the work done by the force field $F(x,y,z)=(y^2,z^2,x^2)$ along the curve of intersection of the sphere $x^2+y^2+z^2=1$, the cylinder $x^2+y^2=x$, and the halfspace $z>0$. The path is ...
1
vote
2answers
55 views

Find $\int\frac{dx}{2+\sqrt{x}}$ (using Integration by Substitution)

I used the substitution: $u=x$ $du=dx$ $2+\sqrt{u}=2+\sqrt{x}$ I then substituted the u into the equation: $\int\frac{1}{2+\sqrt{u}}du$ $=\int{(2+\sqrt{u})^{-1}du}$ I'm not too sure how to ...
9
votes
4answers
322 views

What are other methods to evaluate $\int_0^1 \sqrt{-\ln x} \ \mathrm dx$

$$\int_0^1 \sqrt{-\ln x} dx$$ I'm looking for alternative methods to what I already know (method I have used below) to evaluate this Integral. $$y=-\ln x$$ $$\bbox[8pt, border:1pt solid ...
2
votes
2answers
252 views

Trig substitution fails for evaluating $ \int \frac{\cos x \sin x}{\sin^2{x} + \sin x + 1} dx$?

Evaluate the integral \begin{equation} \int \frac{\cos x \sin x}{\sin^2{x} + \sin x + 1} dx \end{equation} Basically I could substitute: $t = \sin x$ and get: $$\int \frac{t}{t^2 +t + 1} dt$$ But, ...
2
votes
4answers
177 views

Anyone can integrate $e^{-\frac{x^2}{3}}$ by hands?

I just used wolfram integral calculator and the result is weird, there is something called error function. $$ \int_{-\infty}^\infty e^{-\frac{x^2}{3}}\,\mathrm dx $$ Hint says that change of variable ...
2
votes
1answer
55 views

Evaluating $\int_0^\infty dn \, \frac{x^n}{(3n+1)(3n+2)}$

I'm trying to prove a particular series is convergent, and I would like to use the Cauchy integral test for fun, even though it's not the most convenient. I need to evaluate, $$\int_0^\infty dn \, ...
1
vote
5answers
109 views

Integrating $\int_{\sqrt{2}}^2 \frac{1}{t^3\sqrt{t^2-1}}\,dt$.

I am trying to compute $$ \int_{\sqrt{2}}^2 \frac{1}{t^3\sqrt{t^2-1}}\,dt. $$ This is what I got so far: $t=\sec(x)$ and $dt=\sec(x)\tan(x)x\,dx$ So plugging this in gives me $$ \int ...
1
vote
2answers
34 views

Study the convergence of $\int_1^\infty \frac{\arctan x }{x^2}dx$

Study the convergence of $\int_1^\infty \frac{\arctan x }{x^2}dx$ I've seen a proof which goes like this. $$ \lim_{x\to\infty} \frac{\frac{\arctan x}{x^2}}{\frac{1}{x^2}} = \frac{\pi}{2} > ...
1
vote
2answers
46 views

Integration of the following

What is the definite integral of $$ \int_0^1 \left(\frac{g(x)}{f(x)}\right)'\cdot\frac{1}{g(x)}\,dx, $$ where the conditions are as follows: $f(0) = 2 $ $f(1) = 3 $ $f'(x) $ is continuous For all ...
2
votes
1answer
62 views

finding $\int\frac{1}{(t^2+25)^2} dt$ without trig substitution

Our calculus book covers partial fractions but not trig substitution, so I would like to find out the most elementary way to evaluate $$\displaystyle\int\frac{1}{(t^2+25)^2}\;dt$$ without using ...
9
votes
6answers
438 views

Hints on calculating the integral $\int_0^1\frac{x^{19}-1}{\ln x}\,dx$

I would be happy to get some hints on the following integral: $$ \int_0^1\frac{x^{19}-1}{\ln x}\,dx $$
0
votes
1answer
52 views

Evaluate the integral $\int_1^\infty \frac{2^x}{2^{(2^x)}}dx$

Evaluate the integral $$\int_1^\infty \frac{2^x}{2^{(2^x)}}dx$$ My Try: substituting $t = 2^x$ we get: $$\ln 2 \int_2^\infty \left(\frac{1}{2}\right)^t dt = \frac{\ln 2}{\ln 0.5} \left( ...
-3
votes
0answers
36 views

Here is an (QFT, QED related) integration problem in 2+1 D $\vec{k}$ is spacial, $k^{\circ}$ is temporal part. Any suggestions how to proceed? [on hold]

Kindly check it.I don't know what to do with this delta and etc etc $\int \frac {e^{-i\vec{k}.( \vec{x}'-\vec{x})}e^{-ik^{\circ}x_{\circ}}\delta(k^{\circ})}{k^2-\mu^2}d^2kdk^{\circ}$
3
votes
0answers
49 views

$\lim_{n \rightarrow \infty}\frac{1}{n}\sum_1^n\frac{k^8}{(a+(k+b)^2)^4}=1$

I am just having fun with this question: Is this true that $\displaystyle \lim_{n \rightarrow \infty}\frac{1}{n}\sum_1^n\frac{k^8}{(a+(k+b)^2)^4}=1$? I thought to change this to an integral, namely ...
1
vote
1answer
22 views

Sum of uniform random variables $U(0,1)$ and $U(0, a)$

The problem I have is: $X \sim U(0,1), Y \sim U(0,a)$ are independent random variables. Find the pdf of $X + Y$. I've got stuck in an integral-problem, and will show you what I've tried. Skip to the ...
1
vote
0answers
59 views

Volume of figure between $x^2+y^2+z^2=16$ and $ x^2+y^2=6z$ if $z\geq 0$

I have a problem where I have to find volume of figure formed, when $x^2+y^2+z^2=16$ and $ x^2+y^2=6z$ intersects if $z\geq 0$. Here is a graphic for clarity: So far I have transformed the problem to ...
2
votes
2answers
272 views

Find $\lim_{x \to 0}\frac{\int_0^x(e^{2t}+t)^{1/t}dt}{x}$

It's asked to solve this: $$\lim_{x \to 0}\frac{\int_0^x(e^{2t}+t)^{1/t}dt}{x}$$ And I have no idea how to do it...
0
votes
0answers
7 views

Transformation of the infinitessimal integration variable under a coordinate transformation

I always get confused when I'm facing the 3D integral over space and have to do a coordinate transformation on the given function to solve the integral. Do some of you have tips/trick how to ...
-1
votes
0answers
63 views
+50

analytic solution of a definite integral

Integrate the following $$\int_0^\infty \alpha\,\beta\, c\, k\, x^s\, x^{c-1} (1+x^c)^{k-1} \left[(1+x^c)^k-1\right]^{-\beta-1} \left[1+\gamma ...
0
votes
3answers
125 views

Is there any difference at all between the antiderivative and the original function?

A definite integral on the interval $(a,b) =$ the antiderivative at $b -$ the antiderivative at $a$. Am I correct in saying that the antiderivative $=$ the original function? Because the integral is ...
1
vote
1answer
38 views

Why doesn't this method for integration by parts work?

So here is what I did first. $$∫16\ln(x^{1/3})dx$$ move the constant $16$ out $$16∫\ln(x^{1/3})dx$$ use properties of logarithms to rewrite natural log of cube root of $x$ as $\ln x$ divided by $3$ ...
2
votes
3answers
136 views

What's the process that lead me from integral of $\tan(x)$ to $-\ln(\cos(x))$?

I'm trying to compute $\int \tan(x) dx$. I tried to decompose it to $\int\sin(x)\cdot\cos(x)^{-1} dx$ and use per partes method. Then I got stuck at ...
1
vote
1answer
20 views

Find all differentiable equations using Cauchy-Riemann equations

Let $z=x+iy$ and $f(z)=u(x,y)+iv(x,y)$. I want to use the Cauchy Riemann equations to find all differentiable functions of the form $$Re( h(z))=2x^2+2x+1-2y^2$$ So I used the C-R equations with ...
4
votes
1answer
34 views

Check my answer - complex analysis, using residue and rouche's theorem

I was asked the following questions and I am unsure of my solutions, any advice would be appreciated, maybe there is a better way of doing this. Question: We are given $f(z)=2z-\sinh (z)$ defined on ...
1
vote
2answers
44 views

Laplace transform convolution attempt

I can't seem to get this Laplace working using the convolution method. $H(s) = \frac{1}{s^2(s+2)}$ Which I can't get to work using convolution. So I am separating it into $\frac{1}{s^2} * ...
1
vote
1answer
53 views

Fundamental theorem of calculus, differentiable at the endpoints.

One version states: Let f be a continuous real-valued function defined on a closed interval $[a,b]$. Let f be the function defined for all x in $[a,b]$, by $F(x)=\int_{a}^xf(t)dt$. Then, F is ...