Tagged Questions
6
votes
0answers
41 views
How do solve this integral $\int_{-1}^1\frac{1}{\sqrt{1-x^2}}\arctan\frac{11-6\,x}{4\,\sqrt{21}}\mathrm dx$?
I need to solve the to following integral:
$$\int_{-1}^1\frac{1}{\sqrt{1-x^2}}\arctan\frac{11-6\,x}{4\,\sqrt{21}}\mathrm dx.$$
I tried this integral in Mathematica, but it was not able to solve it. ...
6
votes
2answers
53 views
How to solve $\int_0^\infty J_0(x)\ \text{sinc}(\pi\,x)\ e^{-x}\,\mathrm dx$?
I need some help with solving this integral involving Bessel function:
$\hspace{2in}\displaystyle\int_0^\infty$$J_0(x)\ $$\text{sinc}(\pi\,x)\ $$e^{-x}\,\mathrm dx.$
3
votes
0answers
50 views
looking for reference or nice proof of trig lemma
Math people:
I am looking for a reference or a nice proof of the following fact. I have proven it myself, but my proof is messy: let $\theta \in (0,1]$ and $\alpha \in (0, \frac{1}{2}\theta^2]$. ...
0
votes
1answer
46 views
Math word problem. Any help is appreciated.
A math student writes a proof of the derivative of a certain trigonometric function. The last line she writes before stating her conclusion is...
$\dfrac{d}{d\theta} \left(\sin\left(\theta ...
5
votes
4answers
72 views
Integrate ${\sec 4x}$
How do I go about doing this? I try doing it by parts, but it seems to work out wrong:
$\eqalign{
& \int {\sec 4xdx} \cr
& u = \sec 4x \cr
& {{du} \over {dx}} = 4\sec 4x\tan 4x ...
0
votes
1answer
61 views
Why do these trig functions “overpower” each other?
For example, $\sin(x)\cos(x)$ can be written as $\sin(2x)/2$, the limit as $x$ approaches $0$ of $\sin(x)\cos(x)$ is $0$, and the limit as x approaches $\pi/2$ is $0$. I don't see a reason why sine ...
12
votes
3answers
147 views
$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$
Is it possible to calculate the following infinite sum in a closed form? If yes, please point me to the right direction.
$$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$$
3
votes
3answers
102 views
How do you integrate the following trigonometric function involving sin and cos?
How do you integrate the following functions:
$$\int \left( \frac{\cos\theta}{1+\sin^2\theta} \right)^2 \, d\theta$$ and $$\int \left( \frac{\cos\theta}{1+\sin^2\theta} \right)^3 \, d\theta
$$
...
8
votes
1answer
88 views
$\int_0^\infty\text{Ci}(x)^3\mathrm dx$
Is there a closed form for this integral:
$$\int_0^\infty\text{Ci}(x)^3\mathrm dx,$$
where $\text{Ci}(x)=-\int_x^\infty\frac{\cos z}{z}\mathrm dz$ is the cosine integral?
15
votes
4answers
185 views
$\int_0^\pi\frac{3\cos x+\sqrt{8+\cos^2 x}}{\sin x}x\ \mathrm dx$
Please help me to solve this integral:
$$\int_0^\pi\frac{3\cos x+\sqrt{8+\cos^2 x}}{\sin x}x\ \mathrm dx.$$
I managed to calculate an indefinite integral of the left part:
$$\int\frac{\cos x}{\sin ...
7
votes
2answers
138 views
$\cos(x)+\cos(x\sqrt{2})$ is not periodic
Show that the function $$f(x)=\cos(x)+\cos(x\sqrt{2})$$ is not periodic. I tried $x = a$ and $a\sqrt{2}$. I am guessing that the method of contradiction would be of some help over here. What else ...
7
votes
1answer
88 views
Need help with calculating this sum: $\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$
I need help with calculating this sum:
$$\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$$
0
votes
1answer
62 views
How to find a point on the tangent line whos length is 1?
im trying to figure out a formula to find the point(x,y) on a tangent line whos length is between 0 and 1 while it rotates around the unit circle uniformly, so the point would either be right on the ...
12
votes
2answers
118 views
Proving the inequality $\tan(1)\le\sum_{k=1}^{\infty} \frac{\sin(1/k^2)}{\cos^2 (1/(k+1))}$
How am I supposed to prove this inequality?
$$\tan(1)\le\sum_{k=1}^{\infty} \frac{\sin\left(\frac{1}{k^2}\right)}{\cos^2 \left(\frac{1}{k+1}\right)}$$
Jordan inequality might be an option but led me ...
5
votes
1answer
75 views
Find the following integral: [duplicate]
Find $$\int \sqrt{\tan x}dx$$
My attempt:
$$\text{Let}\ I=\int \sqrt{\tan(x)}dx$$
$$\text{Let}\ u=\tan(x), du=(1+\tan^{2}(x))dx$$
$$I=\int \frac{\sqrt{u}}{u^{2}+1}$$
$$\text{Let}\ v=\sqrt{u}, ...
5
votes
1answer
80 views
Interesting definite integral involving exp and trig
I'm trying to evaluate the following integrals:
$$\int_0^{2\pi} e^{\kappa \cos(\phi - \mu)} \cos(\phi) d\phi$$
$$\int_0^{2\pi} e^{\kappa \cos(\phi - \mu)} \sin(\phi) d\phi$$
for which I want to find ...
0
votes
2answers
80 views
Whats the maximum value of $ y=6\cos\left( \frac {2\pi}{14} x\right)-2?$
Please show the correct way how to do this thanks.
I got this..
y will be maximum when cos2pi/14x - 2 is maximum
i. e. when 2pi/14x- 2 = 0 [ cos0 is maximum = 1 ]
so or 2pi/14x = 2
or 14x = pi
...
0
votes
1answer
77 views
Prove this proprety of $f(x)$
I've asked this question before a long time ago, but I didn't get a complete answer. This is the link to the incomplete answer: Prove the following property of $f(x)$?
Let ...
3
votes
4answers
86 views
Trigonometric Formula
I am stuck with the simple expression
$$
\frac{\cos^2(\theta + \alpha)}{1 - \cos^2(\theta - \alpha)} = \text{const.}
$$
where $\theta$ is a variable and $\alpha$ is the number satisfying
$$
\alpha = ...
2
votes
1answer
40 views
Cone shaped related rates of change question
A container is in the shape of a cone of semi-vertical angle $30^\circ $, with it's vertex downwards.
Liquid flows into the container at ${{\sqrt {3\pi } } \over 4}{\rm{ }}c{m^{^3}}/s$
At the ...
2
votes
3answers
95 views
Fractional Trigonometric Integrands
$$∫\frac{a\sin x+b\cos x+c}{d\sin x+e\cos x+f}dx$$
$$∫\frac{a\sin x+b\cos x}{c\sin x+d\cos x}dx$$
$$∫\frac{dx}{a\sin x+\cos x}$$
What are the relations between the numerator in the denominator, and ...
0
votes
3answers
60 views
Differentiate $y = {(x + 2)^3}{(1 - \sin 2x)^2}{(1 + \tan x)^3}$
I haven't got very far in attempting this:
$\eqalign{
& y = {(x + 2)^3}{(1 - \sin 2x)^2}{(1 + \tan x)^3} \cr
& y = {\left( {(x + 2)(1 + \tan x)} \right)^3}{(1 - \sin 2x)^2} \cr} $
I'm ...
9
votes
2answers
172 views
Inequality $\sum_{1\le k\le n}\frac{\sin kx}{k}\ge 0$
Show the following inequality for any $x\in [0, \pi]$ and $n\in \mathbb{N}^*$,
$$
\sum_{1\le k\le n}\frac{\sin kx}{k}\ge 0.
$$
I have this question a very long time ago from a book or magazine but I ...
0
votes
1answer
67 views
0
votes
2answers
26 views
Turning points on $2\sin x - x$
I'm self teaching and doing a book exercise which asks: "Considering only positive values of x, locate the first two turning points on the curve $2\sin x - x$ and determine whether they are maximum or ...
4
votes
5answers
133 views
Integrate $\int_0^\pi{{x\sin x}\over{1+\cos^2x}}dx$.
Integrate $\displaystyle \int \limits_0^\pi{{x\sin x}\over{1+\cos^2x}}dx$. I tried substituting $t=\cos x$, and then integrate with integration by parts. It got all messy... Thanks in advance for any ...
2
votes
2answers
64 views
$x_n$ is the $n$'th positive solution to $x=\tan(x)$. Find $\lim_{n\to\infty}\left(x_n-x_{n-1}\right)$
$x_n$ is the $n$'th positive solution to $x=\tan(x)$. Find $\lim_{n\to\infty}\left(x_n-x_{n-1}\right)$.
2
votes
3answers
63 views
Confusion regarding the derivation of $\cos(x)$ when differentiating $\sin(x)$
The textbook I'm reading derives it like this:
$\eqalign{
& y = \sin x \Rightarrow \left( 1 \right) \cr
& y + \delta y = \sin (x + \delta x) \Rightarrow (2) \cr} $
Subtracting equation ...
1
vote
4answers
61 views
What is $g'(x)$ if $g(x) =x^2 \int_{x-2}^{\sin x} \cos^2t dt $?
What is $g'(x)$ if $$g(x)= x^2 \int_{x-2}^{\sin x} \cos^2t dt?$$
So i get $g'(x) = 2x(\int_{x-2}^{sinx} cos^2t dt ) + x^2(cos^2(sinx)-cos^2(x-2))$ as my final answer. Is this right?, thanks. Use ...
1
vote
1answer
70 views
$\cos$ not a contraction on $\mathbb R$
I know that $\cos$ is a contraction mapping on $[0, a]$ with $a<\pi/2$.
I also know that the proof of this uses the mean value theorem and it fails on $\Bbb R$.
However, this is not a proof to the ...
8
votes
2answers
85 views
How to find the minimum of f(x)?
I need to find the minimum of $f(x)$ with
$$f(x)=(\sin(x)+\cos(x)+\tan(x)+\cot(x)+\sec(x)+\csc(x))^2$$
Could you help me with some clues?
0
votes
1answer
39 views
Maximum and minimum of $y = 4x-8*(\cos(x))$ between $-\pi$ and $\pi$
I have found that the maximum of this function is at $\pi$, where the function will equal $$4\pi+8,$$ which is approximately $20$. However, I tried to get the minimum value, and it was incorrect. The ...
1
vote
2answers
99 views
Solve the integral $\int_{x=0}^{\infty}\frac{1}{x}\int_{y=0}^{x}\frac{\cos{(x-y)}-\cos{x}}{y}dydx$
Moderator Note:
This is problem 17709 from the American Mathematical Monthly. Please do not ask ongoing contest questions. This question will be locked until the end of the submission period ...
7
votes
2answers
120 views
Integrate $2\int x^2\, \sec^2x \,\tan x\, dx$
$$
2\int x^2\, \sec^2x \,\tan x\, \mathrm{d}x
$$
How to solve this using integration by parts? WolframAlpha can solve it, but is unable to give a step-by-step solution, and has a different answer to ...
1
vote
2answers
59 views
Finding an infinite trigonometric sum
Find the following infinite sum : $$q\sin a+q^2\sin 2a+\ldots+q^n\sin na+\ldots$$ where $|q|<1$ .It would be good if you could find it without the help of any auxiliary sequences using only ...
3
votes
3answers
113 views
How can I find the points of intersection between the curves $r=1+\sin\theta$ and $r=1-\sin\theta$?
Find the points of intersection for the curve $r=a(1+\sin\theta)$ and $r=a(1-\sin\theta)$
My book says the answer is $(0,0),(a,0),(a,\pi)$.
However I calculated $ (a,0),(a,\pi),(a,2\pi)$.
0
votes
3answers
58 views
Proofs on equilateral triangles
Let $\Delta$ be the set of all triangles with two equal edges and be inscribed in a circle of radius $R$.
So, how do I show that:
Equilateral triangle in $\Delta$ is maximizing the area?
and
this ...
6
votes
3answers
159 views
Trigonometrical limit $\lim\limits_{ x\to 0 } \frac{\sin x - x\cos x}{x^3}?$
Can you help me solve this without using de l'Hôpital's rule (just using Standard rules):
$$ \lim_{ x\to 0 } \frac{\sin x - x\cos x}{x^3}? $$
4
votes
1answer
87 views
Is this a valid proof of the derivatives of the trigonometric functions?
For the sake of this proof, the trigonometric functions $\cos$ and $\sin$ are defined as the coordinates of a point on the unit circle, rather than any of the modern analytic definitions.
Let $\vec ...
9
votes
3answers
190 views
Being ready to study calculus
Some background: I have a degree in computer science, but the math was limited and this was 10 years ago. High school was way before that. A year ago I relearnt algebra (factoring, solving linear ...
2
votes
4answers
176 views
How do I solve $\int\frac{\cos^2(x)}{\sin(x)}\ dx$ without using Weierstass Substitution?
Every problem that I've put into wolfram alpha lately gives me instructions to substitute $\tan(\frac{x}{2})$, but I haven't been taught how to do that, nor can I understand how it works anyhow...
4
votes
2answers
64 views
Which angle to pick for trigonometric substitution?
first timer on this stack exchange so I apologize if this is the wrong place to ask this question
I was wondering how one is supposed to properly pick an angle when using trig substitution to solve ...
2
votes
1answer
94 views
How to solve $\int\cot^5x\sin^2x\ dx$?
I'm not quite sure how to approach this without it getting extremely messy... and even then, I don't know if it will come out right.
The best I can think of is to use IBP, but neither of those ...
2
votes
1answer
32 views
Show a function is constant in its domain
How can I show, using derivatives, that this function is constant in its domain?
$$
{\arctan(x)}+{\arccos \bigg(\frac x {(1+x^2)^\frac 1 2}\bigg)}
$$
Out of curiosity, is there another way without ...
2
votes
2answers
97 views
$\int_{-\infty}^\infty \frac{\lambda l}{2\pi\epsilon_0(x^2+l^2)^{3/2}}dx$ Proving an electric field from a wire falls off at $1/l$
If you don't know the physics behind all this, that's okay, I just need the integral of this function (or limit, I'm not too sure).
Here's the gist: normally with infinitesimal point charges, there ...
1
vote
2answers
282 views
Find the points on the given curve where the tangent line is horizontal or vertical
$r=e^θ $ $($ Assume $0 ≤ θ ≤ 2π.)$
Apparently I keep getting this answer wrong. I dont know if i need to use $n $ in the answer or not...
2
votes
2answers
69 views
How can this trigonometric inequality related to a limit be proved?
I want to prove that $\;\;\displaystyle \left|\frac{\sin x-x}{x^{2}}\right|\leq\frac{4(\pi/2-1)}{\pi^{2}}\;\;$
for all $x$ such that $x\in\left[0,\pi/2\right]$.
If you look at the graph of the ...
5
votes
0answers
104 views
integral of the product of a trigonometric and an exponential function
Since tan has an odd power I would normally aim to sub $u=\sec(x)$, but I cant get rid of the $2^x$.
$$\int 2^x \tan^9(x^2)\sec(x^2)dx$$
I also tried integrating by parts but it got more complicated. ...
3
votes
1answer
76 views
Need help integrating $\frac{(t-1)^2-2t(t-1)}{t^2+(a(t-1)^2)^2}$
I need to integrate $$a\int_1^2 \frac{(t-1)^2-2t(t-1)}{t^2+(a(t-1)^2)^2} dt$$ $$=a\int_1^2 \frac{-t^2+1}{t^2+(a(t-1)^2)^2} dt $$
$$=-a\int_1^2 \frac{t^2-1}{t^2+(a(t-1)^2)^2} dt $$ with the hint that ...
0
votes
2answers
67 views
Domain of definition of $\frac{x}{\sin(x)}$.
Would the domain of definition of $\displaystyle\frac{x}{\sin(x)}$ be $0$? The function on a graphing calculator looks like it has a lot of infinite limits, so I'm not sure.
