Questions tagged [integration]
For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.
74,658
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Series of nested double integrals
This is kind of a follow-up of my previous question. I'm investigating the following infinite series of nested two-dimensional integrals
$$\sigma(t,t^\prime) = 1 - \int_{t^\prime}^t\mathrm dt_1 \int_{...
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Is this a justified expression for the integral of the floor function?
Mathematica seems to agree with me in general with saying that $\displaystyle\int \lfloor x \rfloor dx = \frac{\lfloor x\rfloor (\lfloor x\rfloor-1)}{2}+\lfloor x\rfloor \{ x \}+C = \frac{\lfloor x\...
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Integral of floor function: $\int \,\left\lfloor\frac{1}{x}\right\rfloor\, dx$
How would you go about solving integral of a floor? The particular problem I have is:
$$\int \,\left\lfloor\frac{1}{x}\right\rfloor\, dx$$
- 3,515
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1
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How can this be re-written with the following identity?
Can this:
$$\frac{\cos x}{4 + \sin^2 x}$$
Be re-written using the fact that:
$$\cot(t) = \frac{\cos (t)}{\sin (t)} = \frac{1}{\tan (t)}$$
I'm not good with algebra, but I'm getting there. I'm ...
- 1,909
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2
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Series of nested integrals
I'm trying to calculate the following series of nested integrals with $\varepsilon(t)$ being a real function.
$$\sigma = 1 + \int\nolimits_{t_0}^t\mathrm dt_1 \, \varepsilon(t_1) + \int_{t_0}^t\...
- 345
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3
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How to integrate $\int\frac{1}{\sqrt{1+x^3}}\mathrm dx$?
In a course, my teacher told us that the following integral is convergent and used the comparison test to prove it; my question is how to find the antiderivative in closed form? It seems to exist; if, ...
- 16.6k
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1
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Power Mean Random Distribution
I'm trying to find a the distribution for the power mean of $n$ random variables on $[0,1]$.
I've got the arithmetic mean: $\frac{n}{(n-1)!}\sum_{k=0}^{\lfloor nx\rfloor}(-1)^k\binom{n}{k}(nx-k)^{n-1}...
- 4,814
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2
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Trig integral $\int ( \cos{x} + \sin{x}\cos{x}) \, dx $
Assume we have:
$$ \int (\cos{x} + \sin{x}\cos{x}) \, dx$$
Two ways to do it:
Use $$\sin{x}\cos{x} = \frac{ \sin{2x} }{2} $$
Then
$$ \int \left(\cos{x} + \frac{\sin{2x}}{2} \right) \, dx = \sin{x} -...
- 9,552
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3
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If $\alpha$ is an acute angle, show that $\int_0^1 \frac{dx}{x^2+2x\cos{\alpha}+1} = \frac{\alpha}{2\sin{\alpha}}.$
If $\alpha$ is an acute angle, show that $\displaystyle \int_0^1 \frac{dx}{x^2+2x\cos{\alpha}+1} = \frac{\alpha}{2\sin{\alpha}}.$
My attempt:
Write $x^2+2x\cos{\alpha}+1 = (x+\cos{\alpha})^2+1-\...
- 1,123
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1
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Stuck at the proof of the Riemann-Lebesgue lemma
I'm currently trying to prove the Riemann-Lebesgue lemma using lower Darboux-sums and an approximation of any integrable function $f: [0,1] \to \mathbb{R}$ defined as
$$t(x) := \begin{cases} m_i & ...
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1
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Trouble deriving DE for fourier transform from DE of function
I am trying to derive an equation which is a standard result in physics (the momentum space Schrödinger equation).
(Background: The wavefunction is a complex valued function of position coordinates ...
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1
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1
answer
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Help with fourier transforms
I am going through a book and having trouble with reproducing some results mentioned. The aim is to solve for $D_{s}$ from equation (1) below
$\int D_{s}(\vec{x}-\vec{a})D_{s}(\vec{y}-\vec{b})Q_{ss}(\...
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Another magic re-write question, picture!
It's the part before and after "Thus".
$$I = \ldots = \int e^{ax} \cos bx \ \mathrm{d}x = \frac{1}{b} e^{ax} \sin bx + \frac{a}{b^{2}} \cos bx - \frac{a^{2}}{b^{2}} I.$$
Thus
$$\left( 1 + \frac{a^{2}...
- 1,909
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Parameterization of an implicit function
I'm trying to find the area of an irregular domain that is bounded by $x = c$, $y = c$, and $c = -A\sin(x/2)\sin(y/2)+\cos(x/2)\cos(y/2)$, where A can vary in the range [-1,1], and x and y are only ...
- 2,509
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2
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Calculating $\displaystyle{\int_0^\infty e^{-i\omega t}dt}$
I was studying Fourier Transform; I could answer to this $$\int_{-\infty}^\infty e^{-i\omega t}dt$$ by Fourier Transform, but I have problem in $$\int_0^\infty e^{-i\omega t}dt.$$ I would be grateful ...
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