Questions tagged [trigonometric-integrals]

Relating to integrations consisting of only(mainly) trigonometric functions and/or requiring substitutions by/of trigonometric functions.

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2
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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: $$ \...
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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 ...
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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 $\int_{0}^{2\pi} \delta(\cos(\theta)-k)d\theta$, $\lvert k \rvert$ <...
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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 ...
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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
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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}} \...
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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) ...
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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....
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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
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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}\...
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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 ))...
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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\...
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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 ...
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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 +...
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1answer
42 views

How to use Lemniscate sine and Lemniscate cosine elliptic integrals?

I’ve 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 ...
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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(\...
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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) \} ...
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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 ...
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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 ...
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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. ...
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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 ...
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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\...
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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
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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 :$...
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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 Huygens–Fresnel principle [1]. Namely, I am attempting to superimpose the secondary wavelets ...
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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
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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
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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
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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 $\...
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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
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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 ...
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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
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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
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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
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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 $\eta$-...
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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
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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 $\mathrm dx=4\sin t\cos t\,\mathrm{d}t$ ...
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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
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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
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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 = \...
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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 ...
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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}...
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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
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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
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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 ...
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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
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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
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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: $$\int \arctan^{3}(x) \sec^{4} (x) dx$$
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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.

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