6
votes
Accepted
Evaluating $\iint\limits_{0\le x\le y\le1}\!\sqrt{1+x^2-y^2}\,{\rm d}x\,{\rm d}y$
Following @PrincessEev's suggestion, use the following change of variables:
$$\begin{cases}x = s\sinh t \\ y = s\cosh t \end{cases} \implies J = \begin{vmatrix}\sinh t & s\cosh t \\ \cosh t & ...
- 36k
5
votes
Integration of derivatives
One way to handle this integral is to just note that
$$
f^2(x)f'(x) = \left(\frac13f^3(x)+C\right)'
$$
so that
$$
\int f^2(x)f'(x)\,dx
= \int \left(\frac13f^3(x)+C\right)' \, dx
= \frac13f^3(x)+C
$$
...
- 27.5k
4
votes
Integration of derivatives
Here is a short response to
from what I've heard, you can't treat dy/dx as a fraction and
canceling them [the dx's] out is not rigorous
...
my question is not about how to solve the integral but ...
- 97.8k
3
votes
Evaluating $\iint\limits_{0\le x\le y\le1}\!\sqrt{1+x^2-y^2}\,{\rm d}x\,{\rm d}y$
Use the integral
$$\int\sqrt{x^2+a^2}dx=\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}2\ln(x+\sqrt{x^2+a^2})+C,$$
\begin{align}
&\iint\limits_{0\le x\le y\le1}\!\sqrt{1+x^2-y^2}\,{\rm d}x\,{\rm d}y\\
=&\...
- 8,525
2
votes
Evaluating the inverse trigonometric integral $\int_{0}^{z}\mathrm{d}t\,\frac{2\arctan{\left(at\right)}}{1+t^{2}}$ in terms of polylogarithms
For $(a,z)\in\mathbb{R}_{>0}^{2}$, we find
$$\begin{align}
\mathcal{J}{\left(a,z\right)}
&=\int_{0}^{z}\mathrm{d}t\,\frac{2\arctan{\left(at\right)}}{1+t^{2}}\\
&=\int_{0}^{z}\mathrm{d}t\,\...
- 30.8k
1
vote
Proof of the General Change of Variables Theorem in Rn?
After a long search I found the demonstration of this theorem, with stronger hypotheses, in the following reference:
[1] Bojarski, Bogdan, and Tadeusz Iwaniec. "Analytical foundations of the ...
1
vote
Solve this integral:$\int_0^\infty\frac{\arctan x}{x(x^2+1)}\mathrm dx$
Let $x=\tan y$, then
$$
\begin{aligned}
\int_0^{\infty} \frac{\arctan x}{x\left(x^2+1\right)} d x & =\int_0^{\frac{\pi}{2}} \frac{y}{\tan y\left(\tan ^2 y+1\right)} \cdot \sec ^2 y d y \\
& =\...
- 22.4k
1
vote
Accepted
Iterative Integration over indicator function of two variables
We do this by considering sections of the function as is done in Fubini's theorem. In this case the function is the indicator.
For $E\in\mathcal{C}\otimes\mathcal{D}$ we define
$$E_{x}:=\{x\in\mathcal{...
- 26
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