Questions tagged [trigonometric-integrals]
Relating to integrations consisting of only(mainly) trigonometric functions and/or requiring substitutions by/of trigonometric functions.
688
questions
2
votes
1answer
84 views
Determining whether $\int_{1}^{+\infty}\frac{\sin^3 \left(x\right)}{\sqrt {x^2}}\,\mathrm{d}x $ converges or diverges.
I's struggling with an integral, and not sure wich method I should use to determining whether it converges or diverges.
I know, from a software, that it should converge.
The integral is:
$$
\...
1
vote
1answer
48 views
An integral involving trigonometric and exponential function
Prove that $$ \int_{0}^{\infty}x^{2019}\sin(\sqrt{3}x)e^{-3x}=\dfrac{2019!\sqrt{3}}{2^{2021}\cdot 3^{1010}} $$
Hence generalize the integral for any other value than $2019$.I know it can be done by ...
0
votes
1answer
33 views
Integrating Dirac delta functions with trigonometric arguments
I am not sure about how to integrate Dirac delta functions which have trigonometric arguments. I am currently trying to work out $\int_{0}^{2\pi} \delta(\cos(\theta)-k)d\theta$, $\lvert k \rvert$ <...
1
vote
1answer
40 views
Unboundedness of ODE with trigonometric coefficients
The task I am working on involves proving that the solutions to the system
$$\dot x = \begin{pmatrix}\cos{t} & \sin{t} \newline \sin{t} & -\cos{t}\end{pmatrix}x$$
are unbounded. To start I ...
0
votes
0answers
22 views
False proof about $\int_{0}^{\infty}\frac{\sin^2(x)}{x^2}dx=\frac{\pi}{2}$
We have :
$$\int_{0}^{\infty}\frac{1}{\cosh(x)}dx=\frac{\pi}{2}$$
There exists a simple antiderivative wich is :
$$2\tan^{-1}\Big(\tanh\Big(\frac{x}{2}\Big)\Big)$$
I recall that :
...
4
votes
0answers
107 views
Solving a difficult integral
I have been stuck on the following integral for some time:
$$
I = \int^{2 \pi}_0 \mathrm{d}\theta \frac{\cos \theta\left(x + \Delta\cos\theta\right)}{\sqrt{k + \left(x + \Delta\cos\theta\right)^2}} \...
0
votes
0answers
44 views
On Integral consisting $\sin^2(f(x))$:
Consider the following integral:
$$I(t)=\int_1^t\sin^2(f(x))dx$$
Here , $f(x)$ is monotonic for the given domain and is at least twice differentiable.
Is there a result (in its full generality) ...
0
votes
2answers
53 views
Definite integral of exponentials and trig functions
Wikipedia has
$$
\int_{-\infty}^{\infty} e^{-ax^2} dx = \sqrt{\frac{\pi}{a}}
$$
and
$$
\int_{-\infty}^{\infty} e^{-ax^2} e^{-2bx} dx= \sqrt{\frac{\pi}{a}} e^{\frac{b^2}{a}}
$$
https://en....
4
votes
2answers
132 views
Integral $I(\tau_1,a,b) = \int_{\tau_1}^\infty d\tau_2\ \frac{1}{b^2 + \tau_2^2} \left(\pi - 2 \tan^{-1} \frac{\tau_2}{a} \right)^2$
I am looking at the integral:
$$I(\tau_1,a,b) = \int_{\tau_1}^\infty d\tau_2\ \frac{1}{b^2 + \tau_2^2} \left(\pi - 2 \tan^{-1} \frac{\tau_2}{a} \right)^2, \tag{1}$$
where $\tau_1$ is real and $a, b$ ...
3
votes
1answer
78 views
Compute $\int \frac{\cos ^2x+\cos (\sin x)}{\sin x \sin (\sin x)+1} \, dx$
From software, I get
$$\int \frac{\cos ^2 x+\cos (\sin x)}{\sin x \sin (\sin x)+1} \, dx=-2 \tan ^{-1}\left(\cos \left(\frac{x}{2}-\frac{\sin x}{2}\right) \csc \left(\frac{x}{2}+\frac{\sin x}{2}\...
1
vote
3answers
77 views
Evaluate $\int_{ 0 }^{ \infty } \frac{ \ln ( x ) }{ { \left( { x }^{ 2 } +1 \right) }^{ 2 } } d x $? [duplicate]
Evaluate:
$$\int_{ 0 }^{ \infty } \frac{ \ln ( x ) }{ { \left( { x }^{ 2 } +1 \right) }^{2}}dx$$
I tried trig substitution, which led me to
$$\int{ \frac{ \ln (\tan ( \theta ))...
0
votes
2answers
23 views
Differential Equations Variations of Parameters and Constant Term
I have a general question about constant terms and trigonometric integrals. The question revolves about why pulling out this $\frac12$ term is important. This is the differential equation I was given\...
1
vote
2answers
47 views
Integral of trigonometric function with parameter
I need to solve the integral $$\int \frac{dx}{1+a\cos x}$$ for $a\>>0$
I tried to use the substitution $t=\tan\frac{x}{2}$ but unfortunately it doesn't seems to work here. after substitude all ...
0
votes
3answers
58 views
Integral rational trigonometric function, what I did wrong?
I would like if someone can tell me what I did wrong. I have the integral : $$\int \frac{dx}{(2+cosx)sinx}$$
This is my solution:
$\begin{align}\int \frac{dx}{(2+cosx)sinx} = \int \frac{dx}{2sinx +...
3
votes
1answer
42 views
How to use Lemniscate sine and Lemniscate cosine elliptic integrals?
Iāve been reading up on lemniscate sine and cosine functions and found that they are essentially the lemniscate analogues of circular sine and cosine functions. The following wiki page elaborates a ...
0
votes
0answers
35 views
Bessel function giving incorrect result
I am trying to reproduce a paper and there is a term in it, $x_{nm}$, which is given as :
$$
x_{nm} = \int_{0}^{\frac{1}{\sqrt{\pi}}} r dr \int_{0}^{2\pi} d\theta \ \psi_{nk_{1}l}^{*} \ r\ cos(\...
0
votes
0answers
9 views
Trigonometric polynomial and Lagrange basis?
Let
\begin{equation*}
\{ t_k \}^{2n-1}_{k=0},\quad t_k = \frac{ k \pi}{n}
\end{equation*}
be collocation points of interval $(0,2\pi) $ and the corresponding Lagrange basis functions $ \{ L_k(t) \} ...
1
vote
3answers
79 views
Calculation of $\int_0^{\frac{\pi}{4}} (\tan(x))^n\,dx$
Evaluate : $$\int_0^{\frac{\pi}{4}} (\tan x)^n \,dx$$
So I tried the following:
Firstly I substituted $\tan x$ as $u$ and tried to convert it to an integrable form or a beta function. That didn't ...
-1
votes
1answer
41 views
Trigonometric integrals problem.
I am trying to solve a stokes theorem verification problem, where i encountered a integral related to trigonometric function. Here is the integral:
$2\int_0^{2\pi}\sqrt{a^2-sin^2t}.(-sint\; dt)$
I ...
1
vote
1answer
18 views
calculate the integral of $\int_D x\sin(y)$
What is $\int_D x\sin(y)\ dA$ where $D$ is the half circle centered at $(0,0)$ with radius $1$ above the $x$ axis.
So I got
$$
\int_0^1 \int_0^\pi r\cos(\theta)\sin(r \sin(\theta))\ r\ dr\ d \theta.
...
1
vote
1answer
34 views
Distribution of $Z = \sin(X) \sin(Y)$ where $X$ and $Y$ are independent and uniform in $[-\pi,\pi]$?
Consider two random variables $X$ and $Y$ that are independent and uniformly distributed over a period, say $[-\pi,\pi]$.
Which is the PDF (or the CDF if you prefer) of $Z = \sin(X) \sin(Y)$?
This ...
2
votes
1answer
84 views
Evaluate $\int\limits_{-1/8}^{1/8}\arccos\left(x^2+5x\right)\ dx$
Find the value of $$\int\limits_{-1/8}^{1/8}\arccos\left(x^2+5x\right)\ dx$$
Answer given
I used property and wrote the integral as $$\dfrac{1}{2}\int\limits_{-1/8}^{1/8} \left( \arccos\left(x^2+5x\...
0
votes
1answer
43 views
Why is this integral wrong?
I don't understand... When I take a definite integral from a to b of 4cos(x)sin(x), u-substitution tells me that the answer should be 2sin^2(x) evaluated from b - it's evaluation at a. But my first ...
3
votes
0answers
42 views
Conjecture about a famous trigonometric integral
I don't know if this conjecture is well know but let me try it :
Let $f(x)$ be a continuous , differentiable function on $[0;+\infty[ $ with $f(x)\geq 0 \quad\forall x\geq 0$ and such that :$...
0
votes
0answers
68 views
What is antiderivative of $f(\theta)$, where $f(\theta) = \cos {( 2\pi \sqrt{ 1 + a^2 } \sec^2(\theta))}$?
CONTEXT
[Note: this post has been altered from its original]
I am working on a problem relating to the HuygensāFresnel principle [1]. Namely, I am attempting to superimpose the secondary wavelets ...
0
votes
1answer
15 views
Vector analysis| Line Integral | Cycloid path
I am given the following to be evaluated along a cycloid from (0,0) to ($\pi$,2)
->now the integral is this:
$$\int_C{(6xy-y^2)}dx+(3x^2-2xy)dy$$
->and the path being a cycloid is this:
x=$\theta$-...
11
votes
4answers
350 views
Evaluate $\int_0^\pi \frac{\sin\frac{21x}{2}}{\sin \frac x2} dx$ (from the MIT Integration Bee)
I recently watched the MIT Integration Bee ($2006$) video and stumbled upon this unusual integral: $$\int_0^\pi \frac{\sin\frac{21x}{2}}{\sin \frac x2} dx$$
I thought multiplying up and down by $\cos \...
1
vote
1answer
20 views
Is the theory of circular functions sufficient to deal with all algebraic, convex simple closed curves of a certain form?
I was trying to evaluate the integral $$\int_0^1\sqrt[4]{1-x^4}\,\mathrm dx$$ but failed to do so in terms of elementary functions. I wondered what happens more generally; that is, for some integer $m&...
2
votes
2answers
126 views
Show $\int_{0}^{\pi} \frac {x dx}{(a^2\sin^2 x+ b^2\cos^2 x)^{2}}=\frac {\pi^2 (a^2+b^2)}{4a^3b^3}$
Show that
$$\int_{0}^{\pi} \frac {x dx}{(a^2\sin^2 x+ b^2\cos^2 x)^{2}}=\frac {\pi^2 (a^2+b^2)}{4a^3b^3}$$
My Attempt:
Let $$I=\int_{0}^{\pi} \frac {x dx}{(a^2\sin^2 x+b^2 \cos^2 x)^2} $$
Using $\...
1
vote
1answer
54 views
Very interesting problem with integral,number theory and irrationality
It's a problem that I found interesting because it gives an approximation of $\frac{\pi}{2}$ and a sequence of term .I can add to my problem a bit of elementary number theory furthermore.Finally and ...
2
votes
1answer
41 views
$\int_{0}^{e}\frac{\sin(\operatorname{W(x)})-1}{\sin(\operatorname{W(x)})+1}\frac{\sin(\operatorname{2W(x)})-1}{\sin(\operatorname{2W(x)})+1}dx=?$
Solve the following integral :
$$\int_{0}^{e}\frac{\sin(\operatorname{W(x)})-1}{\sin(\operatorname{W(x)})+1}\frac{\sin(\operatorname{2W(x)})-1}{\sin(\operatorname{2W(x)})+1}dx=?$$
My attempt:
We ...
0
votes
2answers
33 views
Prove $\sin r = 3\sin \frac r3 - 4\sin^3 \frac r3$
Does anyone know where this identity is explained and proved?
The sine of an angle (specified in radians) can be computed by making use of the approximation $\sin x \approx x$ if $x$ is ...
1
vote
1answer
46 views
Integral inequality$\int_{0}^{e}\operatorname{W^{2^{-n}}(x)}\leq \int_{0}^{e}\operatorname{sin^{2^{-n}}(x)}$
it's problem found with the help of WA :
$$\int_{0}^{e}\operatorname{W^{2^{-n}}(x)}\leq \int_{0}^{e}\operatorname{sin^{2^{-n}}(x)}$$
And $n\geq 2$ a natural number.
$\operatorname{W(x)}$ is the ...
2
votes
1answer
58 views
Integral of $\mathrm{sech}(x)$
I have to take the following integral:
$$\int \mathrm{sech}(x)dx$$
I have decided to do the following substitution:
$$\int \frac{1}{\cosh(x)}dx$$
Then I proceed to the following manipulation:
$$\int ...
19
votes
1answer
606 views
Question about finite analog of $\int_0^\infty \frac{\sin x\sinh x}{\cos (2 x)+\cosh \left(2x \right)}\frac{dx}{x}=\frac{\pi}{8}$
The integral
$$
\int_0^\infty \frac{\sin x\sinh x}{\cos (2 x)+\cosh \left(2x \right)}\frac{dx}{x}=\frac{\pi}{8},
$$
is given as equation $(17)$ in M.L. Glasser, Some integrals of the Dedekind $\eta$-...
0
votes
3answers
34 views
Proving a function defined as an integral is greater than 0
fellow mathematicians.
I was given the following exercise:
Given $x>0$, prove that $F(x)= \int_{0}^{x}{\frac{\sin(t)}{1+t}dt} > 0$
What I am trying to do is the following:
We can obtain the ...
4
votes
1answer
54 views
Why doesn't substitution using $2\sin^2x$ work?
I was solving the integral $\int\sqrt{2x-x^2}\,\mathrm dx$. I divided it as $\int\sqrt{x}\sqrt{2-x}\,\mathrm dx$ and used the substitution for $x=2\sin^2t$ and $\mathrm dx=4\sin t\cos t\,\mathrm{d}t$
...
0
votes
1answer
28 views
$\int_{-1}^{1} \frac{1}{\sqrt{-3u²+4}}du$
How would one go about solving
$$\int_{-1}^{1} \frac{1}{\sqrt{-3u^2+4}}du$$
I was told I need to make a substitution with $u=\frac{2}{\sqrt{3}}\sin(t)$ but I don't see where this substitution comes, ...
10
votes
3answers
326 views
Exact expression of a trigonometric integral
Let $a>2$ be a real number and consider the following integral
$$
I(a)=\int_0^\pi\int_0^\pi \frac{\sin^2(x)\sin^2(y)}{a+\cos(x)+\cos(y)} \mathrm{d}x\,\mathrm{d}y
$$
My question. Does there exist ...
3
votes
6answers
67 views
Trig substitution for $\sqrt{9-x^2}$
I have an integral that trig substitution could be used to simplify.
$$ \int\frac{x^3dx}{\sqrt{9-x^2}} $$
The first step is where I'm not certain I have it correct. I know that, say, $\sin \theta = \...
0
votes
1answer
44 views
Where am I going wrong with the integral $\int\frac{1}{\sqrt{1-x^2}}dx$?
As my question says: Where am I going wrong with this integral?
$$\int\frac{1}{\sqrt{1-x^2}}dx=\int(1-x^2)^{-1/2}dx=\int \frac{u^{-1/2}du}{-2x}=-\frac{1}{2x}2\sqrt{u}=-\frac{\sqrt{1-x^2}}{x}$$
For ...
0
votes
4answers
104 views
Solve indefinite integral $\int\frac{dx}{\sin^2{x}\cos^3{x}}$
I need to solve the integral below
$$
\int\frac{dx}{\sin^2{x}\cos^3{x}}
$$
without using hyperbolic functions but using substitutions like $u=\tan{x}$, $u=\sin{x}$ or $u=\cos{x}$.
Also, I know the ...
0
votes
2answers
55 views
An integral that seemingly has two distinct answers
For the indefinite integral:
$$\int \frac{c}{\sqrt{a^2-x^2}} dx$$
I can use inverse trigonometric functions to obtain two answers:
$c\arcsin\left(\frac{x}{a}\right) +C$ and $-c\arccos\left(\frac{x}{a}...
0
votes
0answers
63 views
How to solve integral $1/(x\cdot \cos(x))$
I'm trying to solve an integral but I'm getting stuck whole the time.
$$\int\frac{2 \sin x}{x^2\cos^2 x}dx$$
I'm not sure if it's totally correct but I've been able to "simplify" the integral to:
$$...
0
votes
0answers
46 views
Nmae of a Famous Integral [duplicate]
Prove that
$$\int\limits_0^\infty\frac{\arctan{x}}{e^{\pi x}-1}dx=\frac{1}{2}(1-\ln{2})$$
I have forgotten the name of this integral. Can someone mention the name here because I am sure that ...
0
votes
2answers
52 views
Improper Integral with Trig as numerator [duplicate]
I've seen a lot of questions on this topic, but they've used concepts/theorems I'm not familiar with. Say I have a function like so:
$f(x) = \int_{1}^{\infty}\frac{\sin(x)}{x^2+1}\,dx$
Without using ...
-1
votes
3answers
53 views
I'm stuck on a trig substitution problem and am absolutely lost.
The problem is
$\int{\sqrt{9x^2+25}\over x}dx$
Right now I've got to this point with all my work:
$x=a\tan\theta$
$3x=5\tan\theta$
$dx={5\over 3} {\sec^2\theta}d\theta$
$$\int{\sqrt{5\tan\theta^...
0
votes
1answer
41 views
Problem with computing $\int{\frac{dx}{2\sin^2 x+3\cos^2 x}}$
How computing this integral:
$$\int{\frac{dx}{2\sin^2 x+3\cos^2 x}}$$
I tried replace $\cos^2 x$ for $1-\sin^2 x$ and solve, but no sucess.
0
votes
1answer
156 views
How to evaluate this integral? $\int \arctan^{3}(x) \sec^{4} (x) dx$ [closed]
im tring to solve this integral, but i have no idea how to do it:
$$\int \arctan^{3}(x) \sec^{4} (x) dx$$
1
vote
2answers
74 views
$\int\frac{1}{(x^2 -8x + 17)^{3/2}}dx$
I changed the denominator to $(x-4)^2+1$, but I'm still struggling to get it to a point where I can integrate.
