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.
74,718
questions
15
votes
1
answer
497
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What is the volume of $\{ (x,y,z) \in \mathbb{R}^3_{\geq 0} |\; \sqrt{x} + \sqrt{y} + \sqrt{z} \leq 1 \}$?
I have to calculate the volume of the set
$$\{ (x,y,z) \in \mathbb{R}^3_{\geq 0} |\; \sqrt{x} + \sqrt{y} + \sqrt{z} \leq 1 \}$$
and I did this by evaluating the integral
$$\int_0^1 \int_0^{(1-\sqrt{...
- 6,724
4
votes
1
answer
903
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Question on the Cauchy principal value integral
Motivated by this wiki page, I put my question here:
How to prove $$\lim_{\varepsilon\rightarrow 0^+} \int\nolimits_a^b \frac{x^2}{x^2+\varepsilon^2} \, \frac{f(x)}{x}dx=p.v.\int_a^b \frac{f(x)}{x}...
user9464
30
votes
10
answers
57k
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Integral of $\frac{1}{(1+x^2)^2}$
I am in the middle of a problem and having trouble integrating the following integral:
$$\int_{-1}^1\frac1{(1+x^2)^2}\mathrm dx$$
I tried doing partial fractions and got:
$$1=A(1+x^2)+B(1+x^2)$$
I ...
user7814
1
vote
2
answers
812
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expectation of incomplete gamma
Is the expectation of the (upper/lower) incomplete gamma function known?
$$\int_0^{+\infty} x \Gamma(A, x) \mathrm dx$$
- 1,511
5
votes
3
answers
186
views
More Computing Integrals
This particular problem has been giving me trouble, and while the math dept tutors did help a great deal, the resulting answer hasn't been accepted by the online homework submission website. Find the ...
- 1,191
1
vote
2
answers
217
views
Is it allowed and if so, how to differentiate this integral?
I have the following expression (everything is $\in \mathbb R$):
$$f(a,b,c)=c\cdot\int_a^b g(t) \cdot h(t,c) \,dt,\quad0\leq a<b$$
I now want to differentiate this function with respect to c: $$\...
- 199
10
votes
3
answers
919
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Integrals $ \int_0^1 \log x \mathrm dx $,$\int_2^\infty \frac{\log x}{x} \mathrm dx $,$\int_0^\infty \frac{1}{1+x^2} \mathrm dx$
I don't get how we're supposed to use analysis to calculate things like:
a)
$$ \int_0^1 \log x \mathrm dx $$
b)
$$\int_2^\infty \frac{\log x}{x} \mathrm dx $$
c)
$$\int_0^\infty \frac{1}{1+x^2} \...
- 2,374
0
votes
2
answers
396
views
Is there a difference between these integral notations?
I've come across these two notations for calculating an indefinite integral but I'm not sure whether or not they are equal:
$f(x)dx$
$\int f(x)dx$
When calculating the indefinite integral, the first ...
- 1,283
3
votes
1
answer
469
views
How to evaluate $\int \frac{\cos(x) - 1}{x^2}\mathrm dx$?
would like a hint with the integral $$\int \frac{\cos(x) - 1}{x^2}\mathrm dx$$Thanks
- 31
1
vote
1
answer
3k
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Using the Fourier integral theorem to evaluate the improper integrals
I'm trying to brush up with Fourier series with Apostol's Mathematical Analysis. I was looking through the Fourier chapter and its Fourier integral theorem. I'm slightly confused on how to approach it ...
- 11
1
vote
1
answer
926
views
Computing the integral of $e^{-x^2}$ over the entire line [duplicate]
Possible Duplicate:
Proving $\\int_{0}^{+\\infty} e^{-x^2} dx = \\frac{\\sqrt \\pi}{2}$
At lunch with a math friend years ago, he showed me an integral whose solution was, he said, so beautiful ...
- 11.6k
3
votes
1
answer
767
views
How do I find the inverse Hankel transform of $k^2e^{-k^2}$?
I am trying to solve:
$$f_l(r)=\int_0^{\infty}e^{-k^2}k^4j_l(kr)dk,$$
where $j_l$ is the spherical Bessel function of the first kind, for any integer l >= 0.
Thanks in advance for any answers!
- 865
1
vote
2
answers
1k
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Function of bounded variation
I'm not exactly sure how an integral might be useful here. Somehow this question I will be asking is supposed to be related to bounded linear functionals but I'm still not seeing how.
Let a function $...
- 4,443
27
votes
4
answers
5k
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Olympiad calculus problem
This problem is from a qualifying round in a Colombian math Olympiad, I thought some time about it but didn't make any progress. It is as follows.
Given a continuous function $f : [0,1] \to \mathbb{R}$...
- 2,727
112
votes
3
answers
33k
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$\int_{-\infty}^{+\infty} e^{-x^2} dx$ with complex analysis
Inspired by this recently closed question, I'm curious whether there's a way to do the Gaussian integral using techniques in complex analysis such as contour integrals.
I am aware of the calculation ...
- 4,014