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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15 views

Is this the correct domain of integration for this double integral, under the following coordinate transformation?

Suppose you had the double integral $\iint \limits_{A} \frac{y^{2}}{x^{4}}e^{xy} \ dx \ dy$, where $A$ is the region defined by $x>0, \ y>0$ satisfying $x^{2} \leq y \leq 2x^{2}, \ \frac{1}{x} ...
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0answers
23 views

Gaussian measures from a finite dimensional space to an infinite dimensional space

It's well known that the characteristic function of a gaussian measure $\mu$ on $\mathbb{R}^d$ is given by: $$ \hat{\mu}(\xi)=\int_{\mathbb{R}^d}exp(i\xi . x)\mu (dx),\qquad \xi \in \mathbb{R}^d $$ ...
2
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1answer
89 views

Evaluate the integral $\int_0^\infty \frac {x^{1/2} dx}{x^2 + 1}$ using method of residues

I am trying to evaluate the integral $\int_0^\infty \frac {x^{1/2} dx}{x^2 + 1}$ using method of residues. I can solve this very easily without the $x^{1/2}$ on top, but I do not know what to do when ...
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0answers
66 views

Contour integral of $\int_{0}^\infty \frac{\sinh(kx)}{\sinh(x)}dx = \frac{1}{2}\tan{\frac{a}{2}}$

From Calculating the Fourier transform of $\frac{\sinh(kx)}{\sinh(x)}$ In the case of zero $\omega$ and integral starts as 0, how do I prove that using contour integral $\int_{0}^\infty ...
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1answer
46 views

Suppose $f : [a, b] \to \mathbb{R}$ is bounded and $f \in \mathcal{R}[c, b]$ for all $a < c < b$. Show $f \in \mathcal{R}[a, b]$.

Question: Suppose $f : [a, b] \to \mathbb{R}$ is bounded and $f \in\mathcal{R}[c, b]$ for all $a < c < b$. Show $f \in \mathcal{R}[a, b]$. where $\mathcal{R}[x,y]$ is the space of ...
2
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1answer
47 views

Laplace transform of $\cos^2(\omega t)$

Find the Laplace Transform of $\cos^2(\omega t)$, where $\omega$ is a constant. From a cosine identity: $cos^2(\omega t) = \frac{1}{2}(1+\cos(2\omega t))$. So then I get: \begin{align} ...
2
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2answers
53 views

Integral of $\frac{1}{1-x}$

I am trying to solve this integral and after a substitution I come to this result $$\int \frac{1}{1-x} \,dx=-\ln |1-x|$$ Now I have the two cases $-\ln (1-x)$ and $-\ln(x-1)$. According to my ...
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0answers
53 views

Show that is $\int_A f$ exists, then so does $\int_A |f|$

Let $f$ be a real-valued function on $A$ in $E^n$. Show that is $\int_A f$ exists, then so does $\int_A |f|$ and $\left | \int_A f\right | \le \int_A |f|$. I am trying to finish Rosenlicht's ...
1
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2answers
69 views

Cauchy-Lipschitz (or Picard-Lindelöf) theorem for Banach spaces?

Usually we meet Cauchy-Lipschitz (or Picard-Lindelöf) theorem while solving ODEs in $\mathbb R^n$. However, now I want to apply this theorem to solve a special evolutionary PDE. I look it up in ...
4
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1answer
37 views

Convert Riemann sum to a definite integral: $\lim_{n\to \infty} \sum_{k=1}^n\sqrt{{1\over n^2}\left(1+{2k\over n}\right)} $

I'm between 2 answers for this question, but I am not sure if either of them are right. $$\lim_{n\to \infty} \sum_{k=1}^n\sqrt{{1\over n^2}\left(1+{2k\over n}\right)} $$ It has to be rewritten as a ...
2
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1answer
143 views

Evaluating $\lim_{n\rightarrow \infty}n\int_{0}^{1}\left(\cos x-\sin x\right)^n\text{ d}x$

$$\lim_{n\rightarrow \infty}n\int_{0}^{1}\left(\cos x-\sin x\right)^n\text{ d}x$$ A) $ \infty$ B) $ 0$ C) $ 1$ D) $ \frac{1}{2}$ E) $\cos 1$ Source: admission 2015 Technical University of ...
6
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2answers
179 views

Closed form of the integral ${\large\int}_0^\infty e^{-x}\prod_{n=1}^\infty\left(1-e^{-24\!\;n\!\;x}\right)dx$

While doing some numerical experiments, I discovered a curious integral that appears to have a simple closed form: $${\large\int}_0^\infty ...
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4answers
41 views

Solving the integral $\int_{-1}^1 2\sqrt{2-2x^2}\,dx$

I'm working on a triple integral and have managed to get it to a certain point: $$\int_{-1}^1 2\sqrt{2-2x^2}dx $$ When I check this with WolframAlpha it gives the answer $\pi\sqrt{2}$, which is the ...
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3answers
42 views

If $f(x)$ is integrable on $[a,b]$ then prove that $f(x-c)$ is integrable on $[a+c,b+c]$.

Okay, So, we have $a,b,c$ on $\mathbb{R}$ with $a < b$. Prove that if $f(x)$ is an integrable function in $[a,b]$ then $f(x-c)$ is integrable on $[a+c, b + c]$ with $\int_a^b f(x) dx$ = ...
1
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2answers
40 views

Convert integral to a series

I have to find an infinitite series expansion for the integral: $$\int \frac{x}{8+x^3} \, dx$$ First, I started by determining the Taylor series of the integrand $$\frac{x}{8+x^3}=\frac{x}{8} ...
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1answer
56 views

Test the convergence of the integral $\int_{-\infty}^{\infty}\frac{e^{-x}}{1+x^2}.$

Test the convergence of the following integral $$\int_{-\infty}^{\infty}\frac{e^{-x}}{1+x^2}.$$I can not find the indefinite integral of the integrand so that we can check at the limits $-\infty$ and ...
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1answer
28 views

Primitive function and integration by parts in $L^{1}(\mathbb{R})$

If $f(x)\in L^1(\mathbb{R})$, $f^{'}(x)\in L^1(\mathbb{R})$, can we conclude that 1) $f(x)=\int^{x}_{a}f^{'}(x)\mathrm{d}x+f(a)$ 2) integrate by parts as ...
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0answers
6 views

Finding the volume bounded by a paraboloid and a cone [duplicate]

Find the volume bounded by a paraboloid $z = 2-x^2-y^2$ and that of a cone $z^2 = x^2 + y^2$. The domain is given by $D = (x,y,z: x^2 + y^2 \le 1. \sqrt{x^2+y^2} \le z \le 2-x^2-y^2)$. Use ...
1
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1answer
28 views

Evaluate this integral by changing it into polar coordinates

Evaluate the integral by changing to polar coordinates $$\int ^1 _{x=0} \int ^x _{y=0} \tan^{-1}(\frac{y}{x})\,dy\,dx$$ What would the limits for the polar coordinates be?
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3answers
111 views

How to solve $y^{\prime}=\sin y$?

I am trying to solve $$y^{\prime}=\sin y$$ I don't know what to do next $\dfrac{dx}{dy} =\dfrac{ dy}{\sin y}$
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0answers
65 views

Contour integral of $\int_\gamma \frac{z}{\sin z}dz$

I hope to evaluate the contour integral of $\displaystyle\int_\gamma \frac{z}{\sin z}dz$ where $\gamma$ is circle of radius $\frac{3\pi}{2}$ centered at $z = 0$ and oriented clockwise. I have ...
1
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0answers
21 views

How do I compute this contour integral involving dirac delta function?

$I = \int_0^\infty \sin{t} \space \delta(t^2 - \frac{\pi^2}{4})dt$ Substituting $t$ as $z$ I try to do the contour integral of a circle around t = 1 in the complex plane. I wonder if I can argue that ...
3
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2answers
94 views

$n$'th Harmonic Number Proof

I couldn't find any simpler explanation of how to prove the n'th harmonic number, so here I am asking... The $n$'th harmonic number is defined as $$H_{n} = \displaystyle\sum_{i=1}^n ...
1
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2answers
66 views

Area between two curves. Split so that it is equal.

There was a question I was doing: Find $y = b$ (find $b$) so that it splits the area between $y=4$ and $y = x^2$ into 2 equal areas. I found the solution which they integrate it to y axis. And ...
2
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2answers
68 views

Why are these three integrals all $0$?

I have the following three triple integrals: $$\iiint_S x \sigma \mathrm{d} V$$ $$\iiint_S y \sigma \mathrm{d} V$$ $$\iiint_S z \sigma \mathrm{d} V$$ where $\sigma = k \left(1 - (\rho / a)^3 ...
4
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1answer
72 views

Integration of Exponential

I am trying to integrate this function $f(x)=e^{-c/x}$. $$\int_{a}^b e^{-c/x} dx \\$$ where $c$ is just a constant and $0<a<b$. But $u$ subsititution leads to me to an integration by parts ...
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0answers
92 views

Intersection of two Jordan-measurable sets.

Let $A\subset \mathbf{R}^{n}$ be a closed rectangle. Define $L$={$C\subset A: C$ is Jordan-measurable}. Prove that if $B,C\in L$ then $B\cup C$, $B\cap C\in L$. Note: my definition of ...
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1answer
35 views

Integration and FTC

Guys I didn't understand how to attack this kind of problem, for example: Given that $$ \int_0^1xf''(2x)dx=\frac{1}{4}\int_0^2xf''(x)dx \ \ (i)$$ and $ f(0) =2, f(2)= 4$ and $ f'(2) =5$, calculate ...
2
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1answer
32 views

On an generalized integral exercise: $ \int_{0}^{+\infty} \frac{dx}{\sqrt{x} | 1-x |^{\alpha}} $.

I am asked to determine for which $\alpha > 0$ does the following generalized integral converge: $$ \int_{0}^{+\infty} \frac{dx}{\sqrt{x} | 1-x |^{\alpha}} $$ I did the following $$ ...
2
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2answers
75 views

Evaluate $\int_a^b(x-a)^3(b-x)^4 dx $

Evaluate $\int_a^b(x-a)^3(b-x)^4 dx $ with $0<a<b$. I've tried the following substitutions : $ y = b - x$ Giving, $\int(b-a-y)^3(y)^4 dy $ Then, $ z = y / (b-a)$ Giving, ...
1
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1answer
39 views

Question on $ L^p $ spaces inequalities to prove limit exists

in my class on real analysis we are currently dealing with $ L^p $ spaces and I have been tackled with this problem from Folland's real analysis stating this: If $ f $ is absolutely continuous on ...
1
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3answers
56 views

Show that the improper integral converges

Let $f[0, \infty)\to\mathbb{R}$, differentiable, positive and $\lim_{x\to\infty} (\log f)'(x) = L < 0$. Prove that $\int_0^\infty f < \infty$. So $\lim_{x\to\infty} (\log f)'(x) = L$ ...
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1answer
50 views

Line integral using Green's Extended Theorem.

Let $C$ be parametrization $\mathbf{r}=\space5\cos(t) \mathbf{i}+\space4\sin(t) \mathbf{j}\space, t \in [0, 2\pi]$. Calculate $\oint_C \mathbf{F}\cdot d \mathbf{r}$ where $F$ is vector field ...
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0answers
23 views

What would the limits of integration be for this double integral?

Suppose you had the double integral $\iint \limits_{A} \frac{y^{2}}{x^{4}}e^{xy}dx \ dy$, where $A$ is the area defined by $x>0, \ y>0, \ x^{2} \leq y \leq 2x^{2}, \ \frac{1}{x^{2}} \leq y \leq ...
1
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1answer
61 views

Evaluating $\int^\infty _{-\infty} \frac{e^{-i p x / h}}{x^2 + a^2}\,\mathrm{d}x $

I'm trying to figure out this integral but cannot figure out the right substitution $$ \int^\infty _{-\infty} \frac{e^{-i p x / h}}{x^2 + a^2}dx $$
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1answer
48 views

Changing the order of integration?

For example if I have: $$\int_0^1 \int_1^2 \int_0^3 f(x,y,z) \space dx \space dy \space dz $$ If I want to change the order of integration e.g. to $( dz \space dy \space dx )$ $$\int_0^3 \int_1^2 ...
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1answer
48 views

Find the formula $\int_0^1f(x)dx\approx A_0f(0)+A_1f(1)$ that is exact for all functions of the form $f(x)=ae^x+b\cos(\pi x/2)$. [closed]

Find the formula $\int_0^1f(x)dx\approx A_0f(0)+A_1f(1)$ that is exact for all functions of the form $f(x)=ae^x+b\cos(\pi x/2)$. I'm not sure how to go about this. Any solutions/hints are greatly ...
0
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1answer
20 views

Double integration (correction)

I need your help to verify my answer. This is an exercice who I have to make but the manual doesn't give the answer. I need to calculate the charge : $\sigma(x,y)=x+y+x^2+y^2$ on the disc : $x^2+y^2 ...
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2answers
52 views

Differentiate an exponential integral

Would you guide me differentiating this integral for $m$: $$\frac{\text{d}}{\text{d}m} \int_{x=-\infty}^m\int_{y=n}^{+\infty}\exp\left(-\left[\left(\frac{x-a}{b}\right)^2 - ...
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2answers
29 views

First order differential equation - area.

a) Solve the differential equation: $$(x+1)\frac{dy}{dx}-3y=(x+1)^4$$ given that $y=16$ and $x=1$, expressing the answer in the form of $y=f(x)$. b) Hence find the area enclosed by the graphs ...
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5answers
54 views

Evaluate the integral $\iint\limits_R \sqrt{1-x^2} {d}A$

If $R = \{(x, y) | -1 \leqslant x \leqslant 1, -2 \leqslant y \leqslant 2 \}$, evaluate the integral $$ \iint\limits_R \sqrt{1-x^2} {d}A $$ The author of the book ('Multivariable Calculus' by James ...
3
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2answers
25 views

Finding the inverse laplace of this function: $ F(s)= \frac{s+8}{s^{2}+4s+5}$

Im trying to find the inverse laplace of : $ F(s)= \frac{s+8}{s^{2}+4s+5}$ I reached the following: $$ F(s)= \frac{s}{(s+2)^{2}+1} + 8 \times \frac{1}{(s+2)^{2}+1}$$ Now i have the 2nd term in the ...
3
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2answers
61 views

I need integrate this $\int_{} \frac{1}{\sqrt{1-z^2}-z} dz$

I don't know how to integrate this $\int_{} \frac{1}{\sqrt{1-z^2}-z} dz$ I tried with suspstitution $ t=\sqrt{1-z^2}-z $ but it doesn't work. Please help!
0
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0answers
33 views

Fourier Transform of a kernel

Let $\omega \in \mathcal{S}(\mathbb{R}^{2})$ and define $u(x) = \int_{\mathbb{R}^{2}}\frac{(x-y)^{\perp}}{|x - y|^{2}}\omega(y)dy$, where $(x-y)^{\perp} = \begin{bmatrix}x_{2} - y_{2}\\-x_{1} + ...
1
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2answers
75 views

Proof that Beta-function $B(m,n)$ = $\frac{n-1}{m}B(m+1,n-1)$?

When m and n are positive integers. It probably has to do with the incomplete Beta-function $B_{sin^2(x)}(m,n)$.
0
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1answer
42 views

How do I calculate the Beta-function $B(m,n) = 2\int_0^{\frac{\pi}{2}}\sin ^{2 m-1}(t) \cos ^{2 n-1}(t)\, dt$

The Beta-Function $$B(m,n) =2\int_0^{\frac{\pi}{2}}\sin ^{2 m-1}(t) \cos ^{2 n-1}(t)\, dt \tag{a}$$ is equal to $$\frac{n-1}{m}B(m-1,n+1) \tag{b}.$$ How do I go from (a) to (b)? (I tried with ...
1
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1answer
41 views

What is the value of $\int_{0}^{2\pi}(x-\pi)^2 (\sin x) dx$?

What is the value of $\int_{0}^{2\pi}(x-\pi)^2 (\sin x) dx$? AFAIK : $f(x)$ is odd function $(x-\pi)^2$ should be even because of square, and it's odd because of $(\sin x)$. Can you explain in ...
0
votes
1answer
28 views

Curve integral, what am I doing?

Suppose $f(x,y)=x^2y^2$ and path defined $\alpha(t)=(\cos t, \sin t)$, $0\leq t\leq 4\pi$ Going to find $\int_{\alpha}f ds$, but not completely sure in the final part. Ok, first $x(t)=\cos t$ and ...
3
votes
2answers
72 views

Evaluate $\int_0^\pi \frac{x}{1+\sin\alpha\sin x} dx$

$$\int_0^\pi \frac{x}{1+\sin \alpha \sin x} dx$$ I need some Hints about how to begin with the problem, because I can't think of anything
0
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1answer
30 views

General technique to check the convergence of an improper integral?

Which of these integrals converge ? I am confused about how to check for the convergence when the functions are more complex inside the integral. My attempt: in option C : integrating gives -2 and ...