All Questions
35
questions
14
votes
1
answer
637
views
Show that $\int_{\arccos(1/4)}^{\pi/2}\arccos(\cos x (2\sin^2x+\sqrt{1+4\sin^4x})) \mathrm dx=\frac{\pi^2}{40}$
There is numerical evidence that $$I=\int_{\arccos(1/4)}^{\pi/2}\arccos\left(\cos x\left(2\sin^2x+\sqrt{1+4\sin^4x}\right)\right)\mathrm dx=\frac{\pi^2}{40}$$
How can this be proved?
I was trying to ...
- 25.7k
23
votes
2
answers
2k
views
Show that $\int_0^{\pi/3}\arccos(2\sin^2 x-\cos x)\mathrm dx=\frac{\pi^2}{5}$
There is numerical evidence that $I=\int_0^{\pi/3}\arccos(2\sin^2 x-\cos x)\mathrm dx=\frac{\pi^2}{5}$. How can this be proved?
I was trying to answer another question, and I got it down to this ...
- 25.7k
18
votes
2
answers
698
views
How to show that $\int_0^\pi \arcsin{\left(\frac{\sin{x}}{\sqrt{5/4-\cos{x}}}\right)}dx=\frac{\pi^2}{4}$?
I am trying to show that
$$\int_0^\pi \arcsin{\left(\frac{\sin{x}}{\sqrt{5/4-\cos{x}}}\right)}dx=\frac{\pi^2}{4}$$
Context: I was working on another question ("Attempt $2$") and miscopied an ...
- 25.7k
9
votes
2
answers
4k
views
Derivative of $\operatorname{arctan2}$
I'm currently working on some navigation equations and I would like to write down the derivative with respect to $x$ of something like $$f(x) = \operatorname{arctan2}(c(x), d(x))$$
I've searched ...
- 117
0
votes
2
answers
495
views
Inverse of $x - \tanh(x)$
I am trying to find out the inverse of function $f:\mathbb R\to\mathbb R, f(x) = x - \tanh(x),\forall\in\mathbb R.$
What I tried:
Since $f(x)$ is invertible, so using $f(f^{-1}(x)) = x,$ I get $x = f^...
0
votes
2
answers
200
views
Is it possible to solve $\cos(x) + 2e^{x} = 0$ analytically?
My Calculus textbook uses $f(x) = \sin(x) + e^{2x}$ as an example of a function with infinitely many local extrema. That much is clear, because $\cos(x) + 2e^{x} =0 $ has infinitely many solutions ...
- 9,692
2
votes
4
answers
126
views
Computing $\int_0^\pi \frac{dx}{1+a^2\cos^2(x)}$
I am trying to compute the following integral
$$\int_0^\pi \frac{dx}{1+a^2\cos^2(x)}$$
But I got stuck on my way.
Indeed, enforcing the change of variables $t =\cos^2x$ leads to
$$\int_0^\pi \...
- 24.2k
10
votes
1
answer
438
views
How to evaluate the integral $\int_0^{\pi/2}x^2(\sin x+\cos x)^3\sqrt{\sin x\cos x} \, dx$?
How to evaluate the integral $$\int_0^{\pi/2}x^2(\sin x+\cos x)^3(\sin x\cos x)^{1/2} \, dx \text{ ?}$$
I tried to subsititution $x=\frac{\pi}{2}-t$, but it doesn't work. can someone
help me, any hint ...
- 1,431
0
votes
1
answer
96
views
Closed form for the recurrence $c_0=-1,\ c_{n+1}=\sqrt{\frac{c_n+1}{2}}$ that computes $\cos(\pi/2^n)$
Let $c_n:=\cos(\pi/2^n)$ for $n\geq 0$. These values can be computed using the following recurrence (I can explain why if needed): $$c_0=-1,\ c_{n+1}=\sqrt{\frac{c_n+1}{2}}.$$
Is there a closed form ...
- 1,602
39
votes
3
answers
4k
views
The entry-level PhD integral: $\int_0^\infty\frac{\sin 3x\sin 4x\sin5x\cos6x}{x\sin^2 x\cosh x}\ dx$
I hope you find this integral interesting.
Evaluate
$$\int_0^\infty\frac{\sin\left(\,3x\,\right)\sin\left(\,4x\,\right)
\sin\left(\,5x\,\right)\cos\left(\,6x\,\right)}{x\,\sin^{2}\left(\,x\,\...
- 11k
2
votes
4
answers
126
views
Help with the integral $\int x\sqrt{\frac{1-x^2}{1+x^2}}dx$
I would like to know what is $$\int x\sqrt{\frac{1-x^2}{1+x^2}}dx.$$ I put $x=\tan(y)$ to get integral of $\displaystyle \int \frac{\sin(y)}{\cos^3(y)}.\sqrt{\cos(2y)}dy$ I don't know whether $\sin(x)...
- 16k
5
votes
2
answers
246
views
A closed form of the series $ \sum_{n=1}^{\infty} q^n \sin(n\alpha) $ [duplicate]
I am having problems with the following series:
$$
\sum_{n=1}^{\infty} q^n \sin(n\alpha), \quad|q| < 1.
$$
No restrictions on $\alpha$. I need to find out whether it converges and if yes, ...
- 93
19
votes
6
answers
1k
views
Need help with $\int_0^\infty\arctan\left(e^{-x}\right)\,\arctan\left(e^{-2x}\right)\,dx$
I was able to calculate:
$$\int_0^\infty\arctan\left(e^{-x}\right)\,dx=G$$
$$\int_0^\infty\arctan^2\left(e^{-x}\right)\,dx=\frac\pi2\,G-\frac78\zeta(3)$$
$G$ is the Catalan constant. In both cases ...
- 3,320
27
votes
3
answers
15k
views
Integral ${\large\int}_0^{\pi/2}\arctan^2\!\left(\frac{\sin x}{\sqrt3+\cos x}\right)dx$
I need to evaluate this integral:
$$I=\int_0^{\pi/2}\arctan^2\!\left(\frac{\sin x}{\sqrt3+\cos x}\right)dx$$
Maple and Mathematica cannot evaluate it in this form.
Its numeric value is
$$I\approx0....
- 5,342
33
votes
3
answers
2k
views
Need help with $\int_0^\pi\arctan^2\left(\frac{\sin x}{2+\cos x}\right)dx$
Please help me to evaluate this integral:
$$\int_0^\pi\arctan^2\left(\frac{\sin x}{2+\cos x}\right)dx$$
Using substitution $x=2\arctan t$ it can be transformed to:
$$\int_0^\infty\frac{2}{1+t^2}\...
- 3,320