# Tagged Questions

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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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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? ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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? ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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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: $$u = 5x^2 - 4x - ... 0answers 21 views ### fourier transform for pde equation I was solving the pde using fourier transform: u_{tt}-u_{xx}+m^2u=0 with initial values u(0,x)=f(x) and u_t(0,x)=g(x). I have received the answer$$U(t,k)=Ae^{-it \sqrt {k^2+m^2}}+Be^{it \sqrt ...
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 ...
### 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 ...