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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29 views

Convolution operator is normal

Consider the convolution operator $$Tf(s)=\frac{1}{2\pi}\int_0^{2\pi}f(t)h(s-t)\,\,dt,\quad f\in L^2[0,2\pi]$$ where $h:\Bbb R\to \Bbb C$ is a $2\pi$-periodic function, square integrable on ...
1
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0answers
38 views

Integral transform involving square root

I am considering the following integral equation $$\frac{1}{y} = \int_a^{\infty} g(x,y) x^{-1/2} \text{ d}x$$ where $g(x,y)$ is to-be-determined and $a$ is a positive constant (if it is instructive, ...
1
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0answers
62 views

Proof that Derivative of Expected Value is Zero (Using Differentiation show Unconditional Expectation is Constant)

If the expected value of a distribution is constant, it means its derivative with respect to the values it can take must be zero. I was wondering if there is a rigorous proof of the same. Steps Tried ...
1
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1answer
50 views

Integral of $x^{-2}e^x$

This is the original problem. $$\int_2^1 \frac{x^2e^x - 2xe^x}{x^4}$$ My attempt at breaking it down $$\frac{x^2e^x}{x^4} - \frac{2xe^x}{x^4}$$ $$x^{-2}e^x - 2x^{-3}e^x$$ $$ ...
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1answer
30 views

Derive the integration formula for the discrete case (Riemann-Stieltjes Integral).

The problem asks to derive the following formula $\sum^n_{k=1} f(k) = [n]f(n)-\int^n_1 f'(x) [x] dx$. Assuming $f'$ is continuous, where $[x]$ is the greatest integer in $x$ I know to use the ...
1
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2answers
39 views

How to find the area for the curve $y=\sin^3(2x)\cos^3(2x)$?

I could calculate the integration of this by substituting $u=\sin(2x)$ and could find one of the limits of integration which was $0$. However, I couldn't find second limit. The mark scheme says the ...
3
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3answers
42 views

What's the point of the fancy notation for surface integrals and line integrals?

Most of the times you see line integrals of a vector field written as this $$ \int_C\mathbf{F\cdot ds} $$ And surface integrals like $$\iint_\Sigma \mathbf{F\cdot n}\,\mathrm dS$$ My question is, ...
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4answers
36 views

What would the value of this integral be when I apply these conditions? (My answer appears wrong for some reason) [on hold]

$$\int\frac{1}{(x+a)(x+b)}\,dx = \left(\frac{\ln|x+a|}{b-a}\right)+\left (\frac{\ln|x+b|}{a-b}\right)\,$$ What would the value of this expression be when $a \ne b$, then what would the value of ...
1
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1answer
39 views

Double integration over function with absolute values

I have having difficulty in how to solve the following double integral problem involving absolute values and the assumption that $\alpha > 1$: $\iint_{-\infty}^{+\infty} \frac{1}{1+|x|^\alpha} ...
0
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0answers
32 views

Is it ok to use $x$ and $-x$ as counterexample for $L_{f+g}(M)\le L_f(M)+L_g(M)$?

Let $P$ be partition of $[a,b]$. How to give an counterexample to $L_{f+g}(P)\le L_f(P)+L_g(P)$ and $U_{f+g}(P)\ge U_f(P)+U_g(P)$? Use $f(x)=x$ and $g(x)=-x$ and restricting domain to $(0, 1]$. In ...
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2answers
105 views
+50

How does a C-constant in a substitution affect the result of integration?

I'm doing some pretty strange integrals (floor functions ones) and I think I should probably start asking some more complex questions regarding it. Since I now know how to integrate them, I have to ...
0
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2answers
42 views

Integral with substitution, how they get to next step

$$2\int\frac{u^2}{u^2-4}\,du = 2\int\left(1+\frac{4}{u^2-4}\right)\,du$$ Source. Can someone please explain how they get from step on the left to the one on the right?
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2answers
38 views

How to prove that if $f$ is integrable, then $\forall \epsilon >0, \ \exists$ partition $M\in [a,b]$ such that $U_f(M) - L_f(M)\lt\epsilon$?

Here is my proof: Since $f$ is integrable, $\overline{I}_a^b(f)$=$\underline{I}_a^b(f)$. However, it is also a fact that $L_f(M) \leqslant \underline{I}_a^b(f) \leqslant \overline{I}_a^b(f) ...
1
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1answer
21 views

How to find proper functions to bound for integral squeeze thereom

I am trying to prove that the function on $[0,1]$ defined by $$f(x)=\begin{cases} 1 & \text{if $x=\frac{1}{n}$ , $n \in \mathbb{N}$}\\ 0 &\text{else} \end{cases}$$ Is Riemann integrable on ...
0
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2answers
42 views

Integral of exponential rational function

I'm asked to find $$\int_0^{\ln 2}{e^{2x}\over{e^{4x}+3}} \text{ d}x$$ I can't for the life of me figure out how to integrate this.
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2answers
61 views

$ \int\frac{\sin(nx) \sin x}{1-\cos x} \,dx$ by elementary methods

What is an elementary way to show that for positive integer $n$ $$ \int\frac{\sin(nx) \sin x}{1-\cos x} \,dx= x + \frac{\sin (nx)}{n} + 2 \sum_{k=1}^{n-1}\frac{\sin(kx)}{k} $$ This cropped up when ...
2
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1answer
34 views

Does $\int f(s) ds = \int g(s) ds \not =0$ imply $f(s)=g(s)$?

specifically for an improper integral, but I'm also wondering about for definite integrals. I'd guess that it's true, but I feel like there must exist different functions that integrate to the same ...
2
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1answer
26 views

Why does the boundary of a region $D$ have enough information to dictate the value of an integral over $D$?

There are many theorems which say something along the lines of the title: The FTC: $\int_a^bf'(x)dx=f(b)-f(a)$. Green's Theorem: Let $F=(P,Q)$, then $\oint_{\partial D}Fds=\iint_D(\frac {\partial ...
3
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1answer
30 views

$\int_{-1}^1\frac1f=\infty$ iff $\int_{-1}^1(u_n')^2f\to0$

Let $f$ be a continuous function on $[-1,1]$ such that $f(x\neq0)>0,f(0)=0$. How can I show that $\int_{-1}^1\frac{1}{f(t)}dt=\infty$ iff there exists a sequence of functions $u_n$, $C^1$ on ...
1
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1answer
89 views

Proving $\int_0^{\pi } f(x) \, \mathrm{d}x = n\pi$

I've been asked to show $$ \displaystyle \int_{0}^{\pi} \dfrac{2(1+\cos x) - \cos((n-1)x) - \cos((n+1)x) - 2\cos nx}{1-\cos 2x} \ dx = n\pi $$ The integrand simplifies nicely to $$\frac{\cos nx - ...
0
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1answer
41 views

Integrating $ x^{\frac{3}{2}} \frac{1}{1 + e^x} $

I'm wondering if this integral can be expressed in some compact form: $$ \int\limits_{0}^{\infty} x^{\frac{3}{2}}\frac{1}{1 + e^x}dx $$ And if not - why? I was thinking that it was somehow ...
0
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2answers
55 views

Find $\int_0^\infty \frac{\sin(4x)}{x}$

How would one go about computing $$\int_0^\infty \frac{\sin(4x)}{x}$$ without any background in complex analysis (e.g. using strictly calculus)? I know that $$\int_0^\infty \frac{\sin(x)}{x} = ...
0
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1answer
31 views

Contour Integration with pole on contour

I have come across an example I don't understand.. So, here is the problematic part: Consider the integrals: $ I = \int_C \frac{e^{iz}}{z} dz $ $ J = \int_C \frac{e^{-iz}}{z} dz $ Where $C,C_-, ...
0
votes
4answers
30 views

u-substitution, indefinite integrals

I've looked on the web for an answer to this question, and could not find an example. Could you push me towards a proper u substitution for the following integral? Please don't solve the problem just ...
0
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0answers
16 views

Choosing a function inside an integral so as to remove the dependence on a particular variable

So I have a slightly strange situation where I am considering: $$\int_t^y \exp\left(-\frac{(g(x)-g(t))^2}{4(x-t)}\right) \frac{ f(y,x)}{\sqrt{x-t}}dx$$ where g is some given function and $f(y,x)$ is ...
2
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2answers
28 views

Simplifying the arc length integral for a segment of a circle (Calc 2)

So I know the length L of the curve $y=\sqrt{R^{2}-x^{2}}$ from $x=0$ to $x=a$ where $|a| < R$ is given by: $$L= \int_0^a \frac{R}{\sqrt{R^{2}-x^{2}}}dx $$ Now I must set up the arc length ...
5
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1answer
58 views

Double Integration over finite plane.

$$\phi(z)=\frac{\sigma}{4\pi\varepsilon_0}\int_{\frac{-a}{2}}^{\frac{a}{2}}\int_{\frac{-a}{2}}^{\frac{a}{2}}\frac{1}{\sqrt{x^2+y^2+z^2}}~dx~dy$$ I'm not sure how to do this integral. For the first ...
0
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3answers
39 views

Symmetry of double integral

I am a little embarassed that I cannot figure out this on my own, but oh well... Let $f \colon [a,b] \to \mathbb{R}$ be continuous. Why is it true that $$ \int_0^1 f(x) \int_0^x f(y) dy \,dx = ...
1
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3answers
56 views

How to integrate $\int_0^{2\pi} \frac12 \sin(t) (1- \cos(t)) \sqrt{\frac12 - \frac12 \cos(t)}\,dt$

How to integrate $$\int_0^{2\pi} \frac12 \sin(t) (1- \cos(t)) \sqrt{\frac12 - \frac12 \cos(t)}\,dt$$ I know the solution is $0$, but I don't know how one gets this.
3
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2answers
89 views

Rigorous proof that $dx dy=r\ dr\ d\theta$

I get the graphic explanation, i.e. that the area $dA$ of the sector's increment can be looked upon as a polar "rectangle" as $dr$ and $d\theta$ are infinitesimal, but how do you prove this ...
0
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3answers
35 views

Setting up this integral?

$$\int\frac{2x+1}{9+x^2}dx$$ I tried to factor out the 9 to get $9(1+\frac{x^2}{9})=9(1+(\frac{x}{3})^2)$ to set up a u-sub to get arctan(x)..... But, it doesn't fit. Is this integration by parts? ...
1
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1answer
72 views

Numerical Integration Error Bound

I would like to use numerical integration to approximate $\int_{0}^{1} f(x) dx$ where $f(x) = \frac{1}{\sqrt{x}}$. But I can't figure out how to get an error bound. For example, if I use trapezoidal ...
5
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1answer
51 views

Continuous but not compact operator on $L^2(0,\infty)$

Define the following operator on $L^2(0,\infty)$: $$Tf(x)=\frac{1}{x} \int_0^xf(y)dy,\quad f\in L^2(0\infty).$$ I would like to see that it is continuous but not compact. So, this is an integral ...
0
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1answer
46 views

Evaluating the integral $ \int \bigl(\bigl(1-\frac{1}{2}z^2\bigr)^{-2}-1\bigr)^{-1/2} dz$ involved in the Young–Laplace equation

Through working on the Young-Laplace equation I cam across the following integral and Maple is acting strange: $$ \int{\frac{1}{\sqrt{\frac{1}{\left(1-\frac{z^2}{2}\right)^2}-1}} \, dz} .$$ If ...
1
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1answer
39 views

Trapezoidal rule or similar for integration over sphere (spherical triangle).

I would like to calculate numerically the integral of the function defined on the sphere. Moreover, the sphere is completely covered by non-overlapping spherical triangles, I need the integral to be ...
0
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1answer
18 views

The function $y=f(x)$ has the property that the chord joining any two points $A(x_1,f(x_1)),B(x_2,f(x_2))$ always intersect $y-$axis at $(0,2x_1x_2)$.

The function $y=f(x)$ has the property that the chord joining any two points $A(x_1,f(x_1)),B(x_2,f(x_2))$ always intersect $y-$axis at $(0,2x_1x_2)$.Given that $f(1)=-1$.Find ...
0
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2answers
41 views

Complex Analysis: Show that $\int_{\gamma}\frac{1}{(z-a)(z-b)}dz=\frac{2\pi i}{a-b}$ [on hold]

How can I show that if $|a|<r<|b|$, then $\int_{\gamma}\frac{1}{(z-a)(z-b)}dz=\frac{2\pi i}{a-b}$, where $\gamma$ is the circle with center the origin, radius $r$, and positive orientation? ...
5
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0answers
92 views

Solve $\int_0^1 \int_0^{2\pi}\frac{ax-x^2\sin(\theta)}{\sqrt{a^2-2ax\sin(\theta)+x^2}}d\theta dx$

Solve $$\int_0^1 \int_0^{2\pi}\frac{ax-x^2\sin(\theta)}{\sqrt{a^2-2ax\sin(\theta)+x^2}}d\theta dx$$ This integral is from the following paper : Frictional coupling between sliding and spinning motion ...
2
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3answers
60 views

How to evaluate these two integrals about hyperbolic functions?

While I was calculating the two integrals below \begin{align*} \mathcal{I}&=\int_{0}^{\infty }\frac{\cos x}{1+\cosh x-\sinh x}\mathrm{d}x\\ \mathcal{J}&=\int_{0}^{\infty }\frac{\sin x}{1+\cosh ...
1
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1answer
93 views

Can this equation be solved without numeric calculation? [on hold]

I want to know the function $f(x)$ which is shown below the integral equation. Can this equation be solved without numeric calculation? Then $\alpha \in \mathbf{R}$ is a scalar. $$ f(x) \cdot \int ...
0
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0answers
23 views

integration of product two lower incomplete gamma function and exponential [on hold]

i need helping to find the value of this integration : $$ \int_0^{\infty}e^{-\delta x}\gamma(\alpha,\theta x){\gamma(\beta,\theta x)\ }dx $$ where all parameters are positive. Can anyone help me how ...
0
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0answers
40 views

How to integrate over inverse of kernel Matrix [on hold]

Let $x_i\in R^m, i \in \{1,2,...,n\}$ be a vector, $x_i(t)=\left( \begin{array}{c} x_{i1}\\ .\\ .\\ x_{ip-1}\\ t\\ x_{ip+1}\\ .\\ .\\ x_{im}\\ \end{array} \right)$, $x_i\neq x_j$ for $i\neq j$ and the ...
2
votes
2answers
76 views

Integral of $x^2 e^{-x^2}$

Like the title says, I'm trying to find $$\int_0^r x^2 e^{-x^2}\,dx$$ Where $r$ is some finite value. I've done one step using integration by parts with $u=x^2$ and $dv=e^{-x^2}dx$, which has left ...
2
votes
1answer
38 views

Integral of a trig function divided by the square root of a polynomial: $\int_a^b\frac{\sin x}{\sqrt{(x-a)(b-x)}}dx$?

I was trying to help some physics students with an integral on their homework and they've presented me with something that has me stumped. The integral they are working on is: $$\int_a^b\frac{\sin ...
0
votes
1answer
68 views

Lower integral of the sum of two functions isn't equal to the lower integral of each summed separately?

I'm trying to figure out the problem above and I know I need to show what I have done so far, but I'm not even sure where to begin.
0
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0answers
20 views

Sum of two random variables - uniform distributions [duplicate]

I have two continuous uniform random variables I need to add. I read that to get the sum of two pdfs you convolve them. I'm getting a bit confused on the limits of integration though. If both RVs are ...
0
votes
2answers
47 views

How do I perform u-substitution on this problem?

I am having trouble with this problem: $$\int {\frac{3x + 5}{5x^2 - 4x - 1}} dx$$ I can't seem to find a u where the du exists in the numerator so that it will cancel. If I choose: $$u = 5x^2 - 4x - ...
2
votes
0answers
21 views

fourier transform for pde equation

I was solving the pde using fourier transform: $u_{tt}-u_{xx}+m^2u=0$ with initial values $u(0,x)=f(x)$ and $u_t(0,x)=g(x)$. I have received the answer $$U(t,k)=Ae^{-it \sqrt {k^2+m^2}}+Be^{it \sqrt ...
0
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1answer
20 views

Prove that for any piecewise smooth curve it is possible to find the parametrisation

Prove that for any piecewise smooth curve it is possible to find the parametrisation $\phi$ that is consistent with its length, ie. length of a curve segment between $\phi(a)$ and $\phi(b)$ is equal ...
0
votes
2answers
82 views

Find all differentiable functions for which $f'(x)+\int_{\pi/4}^x f(t)dt = 0 $

I am having trouble knowing when I have found all possible functions $f(x)$ for the equation. How can I be sure I have found every single one? The question is: Find all differentiable functions ...