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.
74,422
questions
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Substituting $u=\sin(\theta)$ into $\int_{0}^{\pi} \cos\left(\sqrt{1-\sin^2(\theta)}\right) d\theta$ [duplicate]
I would like to show that
$\int_{0}^{\pi} \cos\left(\sqrt{1-\sin^2(\theta)}\right) d\theta = \int_{-1}^{1} \frac{\cos\left(\sqrt{1-u^2}\right)}{\sqrt{1-u^2}} du$ by substituting $u=\sin(\theta)$.
...
- 305
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Leibniz rule when integrand has discontinuity
For $0<y<1$ consider, $$F(y)=\int_0^y \frac{-\log^3 x}{1-x} dx$$
I need to prove that $$F'(y)=\frac{-\log^3 y}{1-y}, 0<y<1$$
By definition $$F'(y)=\lim_{h\to 0}\frac{F(y+h)-F(y)}{h}$$ So ...
- 862
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47
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How to calculate the average distance between two triangles in $\mathbb R^3$?
What I did so far, is recursively subdividing the triangles based on their partial distance, and sum up the distance of the subdivided triangles multiplied by their areas.
This way I only count ...
- 163
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64
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Is there a nice closed form for the integral of $(\tan x)^{2n}$?
I just want to know if there is a nice closed form for this integral:
$$\int_{}^{}(\tan x)^{2n}dx$$
I know that a reduction formula exists and this is what I get by following it:
$$\begin{align*}
\int\...
- 23
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13
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Strict inequality of functions only allows to deduce a non-strict inequality of the expected value of said function
In a proof of Jensen's inequality that I am reading, the following is used: If for a real valued random variable $X$, we have $X(\omega)<\beta$, then $\mathbb{E}[X]\leq \beta$. Why can we deduce ...
- 353
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10
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Expected value of the gradient of logistic loss function to a normal distribution
Given a vector $\beta\in\mathbb{R}^d$, and a random vector $x\sim\mathbf{N}(0_d,I_{d\times d})$, that is $\{x_j\}_{j=1}^d$ are i.i.d generated from gaussian $\mathbf{N}(0,1)$, can we compute or ...
- 1
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29
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Reference for integral of an exponential with quartic argument [closed]
In wikipedia https://en.wikipedia.org/wiki/List_of_integrals_of_exponential_functions
there is this formula.
$$ \int _{-\infty }^{\infty }e^{ax^{4}+bx^{3}+cx^{2}+dx+f}\,dx=e^{f}\sum _{n,m,p=0}^{\infty ...
- 11
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How to calculate $\int_{0}^{\infty} (\frac{1}{2}-\frac{1}{t}+\frac{1}{e^t-1})\frac{e^{-t}}{t}dt$?
I’m reading George E. Andrews’ Special Functions, and it says the integral
$$
\int_{0}^{\infty}\left(\frac{1}{2} - \frac{1}{t} +
\frac{1}{{\rm e}^{t} - 1}\right)\frac{{\rm e}^{-t}}{t}\,{\rm d}t
$$
can ...
- 391
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A Problem with two programmers working on a program, one after another
Problem:
A and B are two computer programmers. They need to write a program. Either one could complete the program in 10 hours working alone. Unfortunately, they only have 1 computer available, so ...
- 4,044
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1
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43
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how do you change the order of integration? [closed]
say you need to integrate f(x,y)dydx,
such as 1≤x≤2,(2−x)≤y≤sqrt(2x−x^2).
how do i change that to dxdy form?
1
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Mellin transform of exponential and logarithm
I am trying to calculate the Mellin transform of the function $f(x) = e^{-ax}\ln\left(1+x\right)$. The mathematica gives me the answer
$$\int_{0}^{\infty}x^{s-1}f(x)dx = \frac{G_{3,5}^{5,2}\left(\frac{...
- 3,263
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$\int_1\dots\int_n\tan^{-1}(x)dx_{1\dots}dx_n$ (Integrating $\arctan$ an Arbitrary Amount of Times)
$$\int_1\dots\int_n\tan^{-1}(x)dx_{1\dots}dx_n$$
I have been trying to solve the above integral, for no reason other than my enjoyment. I am not sure if a closed-form solution exist, or if I am even ...
- 667
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Determine whether $\{\int_0^1 f^2(t)dt = 0\}$ is a subspace of $C([0,1],\mathbb{C})$, complex-valued case
I am currently self-studying through a text on linear algebra, and one of the problems asks to determine whether the set $\mathcal{U} = \{f \in C([0, 1],\mathbb{C}) \colon \int_0^1 f^2(t)dt = 0\}$ is ...
- 43
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Taylor expansion of $\int_{0}^{\omega_0}\frac{\sin\left(\frac{2N+1}{2}\omega\right)\cos(\omega n)}{\sin\left(\omega/2\right)}d\omega$ and similar
I would like to compute the expansion of the following integrals near $N = + \infty$ up to $\mathcal{O}(1/N^2)$: $$ \int_{0}^{\omega_0} \frac{\sin \left( \frac{2 N + 1}{2} \omega \right) \cos(\omega n)...
-1
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This integral converges? $\int_{0}^{\infty }e^{-\Gamma(t)}dt $ [closed]
I would like to know if the following integral is convergent, or divergent:
$ \int_{0}^{\infty} e^{-\Gamma(t)} \rm {d}t$.
