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.
74,712
questions
5
votes
1
answer
223
views
Is this a known special function?
Is this a known special function:
$$\int\nolimits_0^1 a^p(1-a)^{1-p}\\,b^{1-p}\\,(1-b)^p dp\qquad ?$$
I am really only interested in maximizing this over $(a,b)$ in $[0,1] \times [0,1]$, so a ...
steven.p
4
votes
3
answers
697
views
Limit of integral - part 2
Inspired by the recent post "Limit of integral", I propose the following problem (hoping it will not turn out to be too easy). Suppose that $g:[0,1] \times [0,1] \to {\bf R}$ is continuous in both ...
- 24.2k
2
votes
1
answer
617
views
Limit of integral
Let $g: \mathbb{C}\times[a,b]\to\mathbb{R}$ a continuous function and
$$g(t)=g(h_0,t)=\lim\nolimits_{h\to h_0} g(h,t) \ \forall t\in \mathbb{R}$$
Is the following result/reasoning correct ? If we ...
- 4,125
119
votes
13
answers
17k
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Calculating the integral $\int_0^\infty \frac{\cos x}{1+x^2}\, \mathrm{d}x$ without using complex analysis
Suppose that we do not know anything about the complex analysis (numbers). In this case, how to calculate the following integral in closed form?
$$\int_0^\infty\frac{\cos x}{1+x^2}\,\mathrm{d}x$$
- 6,908
2
votes
1
answer
209
views
Liner integral?
Can someone simply explain to me how to calculate linear integral linke below?
$$\int_{L} 5y \mathrm{d}L$$
Where L is line segment from (0;0) to (0,2;0,2).
- 121
2
votes
1
answer
713
views
Can't Solve an Integral
According to the solution manual:
$\int \frac{x}{\sqrt{1-x^{4}}}dx = \frac{1}{2}\arcsin x^{2}+C$
My solution doesn't seem to be working. I know another way of solving it (setting $u=x^{2}$) but the ...
- 855
219
votes
21
answers
160k
views
Evaluation of Gaussian integral $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx$
How to prove
$$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$
- 8,038
0
votes
2
answers
242
views
Proving an Integral
The table of integrals says that
\begin{equation*}
\int \frac{dx}{a^{2}+x^{2}}=\frac{1}{a}\arctan\frac{x}{a}+C
\end{equation*}
where $C$ is a constant. What's wrong with my proof?
$$
\begin{align*}
...
- 855
3
votes
0
answers
334
views
Is this the right way to evaluate this integral?
I am trying to evaluate this integral or at least get bounds for its absolute value.
I have where $\tau \to \infty$
$$\int\nolimits_{1}^{\infty} f(t) \frac{\tau \sin(\tau\log t)}{t^{\sigma+1}} dt$$
...
- 1,913
6
votes
2
answers
7k
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Integrals of the square root of a cubic polynomial
Say I have a function
$$V(x)=A(x-x_1)(x-x_2)(x-x_3)$$
where $x_1$, $x_2$, $x_3$ are the three roots in increasing order and $A$ is positive. Clearly $V(x)$ is positive at large $x > x_3$, ...
Basu
8
votes
6
answers
549
views
Compute $\lim\limits_{a \to 0^+} \left(a \int_1^{\infty} e^{-ax}\cos \left(\frac{2\pi}{1+x^{2}} \right)\,\mathrm dx\right)$
How can I compute the following limit?
$$\lim_{a \to 0^+} \left(a \int_{1}^{\infty} e^{-ax}\cos \left(\frac{2\pi}{1+x^{2}} \right)\,\mathrm dx\right)$$
Any hints you can please give?
Cheers
- 1,265
8
votes
10
answers
13k
views
Evaluate $\int \frac{1}{\sin x\cos x} dx $
Question: How to evaluate $\displaystyle \int \frac{1}{\sin x\cos x} dx $
I know that the correct answer can be obtained by doing:
$\displaystyle\frac{1}{\sin x\cos x} = \frac{\sin^2(x)}{\sin x\cos x}...
- 91
7
votes
1
answer
561
views
Evaluating the integral $\int\limits_{0}^{\infty} \Bigl\lfloor{\frac{n}{e^{x}}\Bigr\rfloor} \ dx $
How to evaluate this integral: $$\int_{0}^{\infty} \biggl\lfloor{\frac{n}{e^{x}}\biggr\rfloor} \ dx, $$where $n \in \mathbb{N}$.
The same integral when asked to evaluate for $n=2$ (say) i can do it ...
anonymous
17
votes
2
answers
74k
views
Volume bounded by cylinders $x^2 + y^2 = r^2$ and $z^2 + y^2 = r^2$
I am having trouble expressing the titular question as iterated integrals over a given region. I have tried narrowing down the problem, and have concluded that the simplest way to approach this is to ...
- 359
10
votes
1
answer
6k
views
Derivative commuting over integral
Can a derivative operation commute over an integral operation irrespective of the properties of the function under the integral ?
- 4,247