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.
74,711
questions
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Integration of $\cos(2x)\cos(nx)$
I'm struggling to integrate $\int \cos(2x)\cos(nx)\,\mathrm{ d}x$
I seem to be going round in circles and would be grateful if someone could help? I think I need to use a trig expansion or identity ...
- 61
2
votes
1
answer
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Integral with analytical solution with normal distribution
I received very good answers a couple of days ago in a simpler related problem, see Integral with Normal Distributions, but I am struggling with this new question:
Let's define a function $F(\theta)=\...
- 292
2
votes
1
answer
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Integral with a natural logarithm in the denominator
How do I solve this?
$$\int \frac{\text{d}x}{x^2 + x \ln x}$$
- 41
5
votes
2
answers
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Questions about Fubini's theorem
I was wondering what theorem(s)
makes possible exchanging the order
of Lebesgue integrals, for instance,
in the following example:
$$\int\nolimits_0^1 \int_0^x \quad 1 \quad dy
dx = \int_0^1 \int_y^1 ...
- 47.7k
5
votes
3
answers
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Integral with Normal Distributions
I know that the following equality is true for any $a$ and $\sigma$ (I have solved it numerically):
$$\int\nolimits_{-\infty}^{+\infty}\Phi\left(\frac{a-x}{\sigma}\right)\frac1{\sigma} \phi\left(\...
- 292
3
votes
1
answer
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Is there a closed-form expression for $\int (a-b\ln(cx))^{-1} \mathrm{d}x$?
Is there a closed form expression for
$$\int\frac{1}{a-b\ln(cx)}\,\mathrm dx\ ?$$
I was wondering how to integrate the above function. I have spent a lot of time on it. First i did an integration by ...
- 365
7
votes
1
answer
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Dyson series and T product
One of the most important tool in quantum mechanics is the Dyson series because it is the basis of the perturbative theory. There is a step in the derivation that I can't understand.
$\{H(t_i)\}$ are ...
- 233
4
votes
1
answer
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How we get the area by subtracting two end points of a function in Integration?
For example consider the following integration:
f(x) = x^3 [from 1 to 3]
$$\int_{1}^{3}x^{3}dx$$
when we subtract: {(3^4)/4}-{(1^4)/4}
why we get the result?
I meant to say, how we get the area ...
- 41
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
$$\int_0^1 \int_0^{(1-\sqrt{...
- 6,724
4
votes
1
answer
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Question on the Cauchy principal value integral
Motivated by this wiki page, I put my question here:
How to prove $$\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
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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
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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
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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: $$\...
- 199
10
votes
3
answers
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Integrals $ \int_0^1 \log x \mathrm dx $,$\int_2^\infty \frac{\log x}{x} \mathrm dx $,$\int_0^\infty \frac{1}{1+x^2} \mathrm dx$
I don't get how we're supposed to use analysis to calculate things like:
a)
$$ \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