Questions tagged [trigonometric-integrals]
Relating to integrations consisting of only(mainly) trigonometric functions and/or requiring substitutions by/of trigonometric functions.
1,227
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Is there a simpler method to compute $\int_0^{\infty} \frac{1}{(1+x)\left(\pi+\ln ^{2n } x\right)} d x$
When I encountered the integral $\int_0^{\infty} \frac{1}{(1+x)\left(\pi+\ln ^{2 } x\right)} d x $, I tried the substitution $x\mapsto \frac{1}{x} $ and found a wonderful result.
$$I=\int_0^{\infty} \...
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Can neglecting the use of a definite integral property when it is very obvious yield incorrect answers? [duplicate]
I was analyzing this integral which prompted me to ask this question:
\begin{aligned}
& \int_0^{2 \pi} \cos ^{-1}\left(\frac{1-\tan ^2 \frac{x}{2}}{1+\tan ^2 \frac{x}{2}}\right) d x \\
& \...
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2
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For $n\ge m\ge 1$, how far can we walk with $ \int_0^{\frac{\pi}{2}} \frac{x^n}{\sin^m x} d x$?
In the post, I tackled the integral by power series and integration by parts and obtained that
$$
\int_0^{\frac{\pi}{2}} \frac{x^2}{\sin x} d x=2\pi G-\frac{7}{2}\zeta(3)
$$
where $G$ is the Catalan’s ...
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Discovering $I_n=\int_0^{\frac{\pi}{2}} \frac{1}{(1+\tan x)^{4n+2}}$ is rational.
Let’s start with the easy one.
$$
\begin{aligned}I_2& =\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{1}{\left[1+\tan \left(\frac{\pi}{4}-x\right)\right]^2} d x \\
& =\frac{1}{4} \int_{-\frac{\pi}...
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votes
3
answers
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does different domain for inverse trigonometric function give different definite integral?
$\int_{-1}^1 {\frac{\tan^{-1}x}{1+x^2}} \; dx$
becomes
$\int_{\tan^{-1}-1}^{\tan^{-1}1} {\theta} \; d\theta$
by substituting $x = \tan(\theta)$
different domain
$\left(-\frac\pi2, \frac\pi2\right)$ or
...
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1
answer
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Generalization of the result of $\int_0^{\infty} \frac{e^{-x^2} \sin \left(x^2\right)}{x^2} d x$.
When I came across the integral $$\int_0^{\infty} \frac{e^{-x^2} \sin \left(x^2\right)}{x^2} d x,$$
I didn’t know how to deal with it. After struggling, I thought of Feynman’s trick and Euler formula ...
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0
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Evaluate the following trigonometric integral with exponential function
Find the value of $$\int_0^\pi\mathrm{e}^{\mathrm{e}^{\cos\left(x\right)}}\cos\left(\sin\left(x\right)\right)\cos\left(\mathrm{e}^x\sin\left(\sin\left(x\right)\right)\right)dx$$
How to solve this ...
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5
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Is there any other method to compute $\int_0^{\frac{\pi}{2}} \frac{x}{\sec x+\csc x} d x$?
Rationalization sometimes makes our life easier
Letting $x\mapsto \frac{\pi}{2}-x$ transforms the integral to
$\displaystyle I=\frac{\pi}{4} \int_0^{\frac{\pi}{2}} \frac{1}{\sec x+\csc x} d x=\frac{\...
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Show $\int_{-\infty}^\infty\int_0^\infty \exp(\cos t-1-t^2) \cos(\sin t-t x)\,\mathrm dt\mathrm dx=\pi$
I am tring to prove
$$
\int_{-\infty}^\infty\int_0^\infty \exp(\cos t-1-t^2) \cos(\sin t-t x)\,\mathrm dt\mathrm dx=\pi.
$$
Numerical integration in Mathematica (truncating the integration bounds on $...
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2
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Evaluate the integral $\int \frac{\cos^2(x)}{1-\beta \sin(x)}\mathrm{d}x$
Evaluate the integral $\int \frac{\cos^2(x)}{1-\beta \sin(x)}\mathrm{d}x$
Using Wolfram, I get a very complex result. Obviously the integration for
$$
\int \frac{\cos^2(x)}{1-\sin(x)} \mathrm{d}x= x-\...
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0
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Madhav-Gregory series and Rudimentary Calculation of PI
I have an equation for the value of PI from the Madhava-Gregory series (but with an ability to make it converge faster). It is quite an obvious equation. But I couldn't find these algorithms on the ...
2
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1
answer
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Simplified expression of $_2F_1(a,a;1+a;-1)$, $a<1$
I need a closed form for the Hypergeometric series $$_2F_1(a,a;1+a;-1)$$ where $a<1$
I tried the following integral representation $$_2F_1(p,q;r;z)=\frac{\Gamma(r)}{\Gamma(q)\Gamma(r-q)} \int_{0}^{...
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0
answers
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discrete least squares error with trigonometric polynomial
Hi I am studying about Trigonometric polynomial that minimize Discrete Least Squares Approximation.
I found a related lecture note online but I got lost in one of the steps on page 27.
I still not ...
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prove orthonormal with respect to discrete inner product
I'm currently reviewing my textbook and I'm having trouble understanding a part here. While it seems intuitively true, I find it hard to prove them in the first place. I've tried working through the ...
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How to make an estimate for this integral?
Suppose that function $f$ satisfies following properties: $f'' \in \mathrm{C}[a,b]$, $\exists A,B\geq 1$ such that $f''(x)\geq 1/A$ and $|f'(x)|<D$ for all $x\in [a,b]$. Prove that $$\displaystyle \...
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How to evaluate $\int_0^{2\pi} [|\sin(x)| + |\cos(x)|]\ dx $ where $[.]$ denotes greatest integer function
Today I came across the following problem:
Evaluate$$\displaystyle\int_0^{2\pi} [|\sin(x)| + |\cos(x)|]\ dx $$ where $[.]$ denotes greatest integer function.
I'm a little confused in simplifying the ...
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6
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Solving $\int\frac{dx}{1+\tan x}$
How do we solve $$\int\frac{dx}{1+\tan x}$$? I found two ways to solve it, one with the Weierstrass substitution ($t=\tan(\dfrac x2)$) and one with simply substituting $u=\tan x$ (which is better in ...
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2
votes
3
answers
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Integral with cosine in the denominator and undefined boundaries
I am trying to solve the integral
$$ \int_0^{\frac{3}{2}\pi}\frac{1}{\frac{5}{2}+\cos(2x)}dx $$
And I get the primitive function to be
$$ \frac{2\arctan\left(\sqrt{\frac{7}{3}}\tan{x}\right)}{\...
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2
answers
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After trigonometric substitution, writing the antiderivative in terms of $x.$
The following integral suggests trigonometric substitution $x=4 \sin (\theta)$ :
$$
\int \frac{x^2}{\left(16-x^2\right)^{3 / 2}} d x \text {. }
$$
After making this substitution and integrating, we ...
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votes
1
answer
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Make an appropriate trigonometric substitution to rewrite the given integrand as an integrand in the angle $\theta$ containing no square roots. [duplicate]
Assume all trigonometric functions are positive.
Remove all square roots.
Do NOT Evaluate the integral completely.
My integral is:
$$\int \frac{\sqrt{49+x^2}}{x^2}\,dx$$
The only substitution I'm ...
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0
votes
0
answers
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Arc length of the graph of $y=\tan x$ on its qurter-period
The arc length of the graph of $y=\sin x$ on its quarter period can be expressed in terms of the gamma function and $\pi$:
$$\int_0^{\pi/2}\sqrt{1+\cos^2 x} \mathrm dx=\frac{\sqrt{2\pi^3}}{\Gamma (1/4)...
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0
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Integration by trig sub $ x=a\sec(t)$. How does it work for $ x=0$?
A question in my textbook says to evaluate
$$\int \frac{dx}{\sqrt{x^2-a^2}}$$
where $a>0$.
I know how to solve the integral using trig substitution but what I do not understand is that my textbook ...
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0
votes
0
answers
29
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Finite Interval of Existence of Solution of ODE [duplicate]
I am struggling on how to prove that the interval of existence of a solution of an ODE is finite. For example, for $x'(t)=x^2+t^2$ where $x \in \mathbb{R}$, I have separated as follows $$\int\frac{1}{...
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1
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Unable to crack $\int_0^{\frac1{\sqrt3}} \frac{\cot^{-1}\sqrt{2-x^2}}{1+x^2}dx=\frac{\pi^2}{30}$
I am unable to solve the integral
$$\int_0^{\frac1{\sqrt3}} \frac{\cot^{-1}\sqrt{2-x^2}}{1+x^2}dx=\frac{\pi^2}{30} $$
after a number of attempts, except with a few observations below.
1). Despite the ...
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2
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Evaluate $\int_0^{\frac{1}{2}}(1-2 x) \ln (\tan (x)) d x$
Evaluate $$\int_0^{\frac{1}{2}}(1-2 x) \ln (\tan (x)) d x$$
My try is using Feyman's Trick:
$$\begin{aligned}
& f(a)=\int_0^{\frac{1}{2}}(1-2 x) \ln (\tan a x) d x \\
& f^{\prime}(a)=\int_0^{...
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1
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Is there any benefit to solving integrals by trig substitution this way?
Suppose we have some sort of integral like $\int\frac{1}{x^2\sqrt{x^2+4}}$ which we can solve by making the substitution $x = 2\tan(\theta)$. After performing the calculations one obtains $-\frac{\csc(...
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Which number is greater A or B?
Let :
$$I_k=\int_{0}^{1}\left(\prod_{n=1}^{k}\left(1+\arctan\left(\left(\frac{y}{4n^{2}}\right)\right)\right)\right)dy$$
And :
$$h\left(x\right)=\int_{0}^{1}\left(\prod_{n=1}^{\operatorname{floor}\...
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1
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Bessel function of the first kind of order zero in integral representation.
I'm studying alternative methods for elliptic boundary conditions. I picked the formula of Bessel function from this site https://dlmf.nist.gov/10.9 I'm looking for any available approach to solve ...
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votes
0
answers
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Half angle sine substitution?
After learning the Weierstrass substitution, I wondered what would happen if I tried doing a half angle sine substitution. Using the formula $$\sin\frac{x}{2}=\pm\sqrt{\frac{1-\cos x}{2}}$$I derived ...
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0
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What is $\int_0^\infty\frac{\sin(\sin(x))}{x}dx$ [duplicate]
So I saw a post concerning the limit of $\frac{\sin(\sin x)}{x}$ as $x\rightarrow0$. So I graphed the function and I wondered what is its integral from $0$ to $\infty$? It seems like it is $\frac{\pi}...
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Integral of $\int_{-\pi}^{\pi} dx e^{-2\pi i (a\cos(x)+b\sin(x))} \sin(mx)$ in terms of standard functions?
Could someone tell me if my derivation of the integral $\int_{-\pi}^{\pi} dx e^{-2\pi i (a\cos(x)+b\sin(x))} \sin(mx)$ is correct. Here $m\in\mathbb{Z}$ (set of integers) and ${a,b}\in\mathbb{R}$ (set ...
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1
answer
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Cosine integral $\int_0^a \frac{1}{{\cos x}\ {\cos (a-x)}}{d}x$, real and complex methods
Evaluate
$$\int_0^a \frac{1}{{\cos x}\ {\cos (a-x)}}\mathrm{d}x$$
One solution begins by expanding the $\cos(a-x)$ term with the addition formula, dividing through by $\cos^2 x$ and substituting $u=\...
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Integral of $\arccos$ function combined with exponontial function
Is it possible to calculate this integral?
$$\int_0^1 x^{j}\ e^{ax^2}\arccos(x)\ dx,\qquad a\in\mathbb{R}_+,\ j\in\mathbb{N}. $$
Although the integral looks neat and fairly simple in form. I tried ...
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Integral of $\,\tan^{2n}(x)\,\mathrm dx$
I want to evaluate $\,\displaystyle I_{n}=\int_{0}^{\frac{\pi}{4}} \tan^{2n}(x)\,\mathrm dx$.
I proved that $\,I_{n}+I_{n-1}=\dfrac{1}{2n-1}\,,\,$ where $I_{0}=\dfrac{\pi}{4}$.
From that I found that (...
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Fourier integral representation of the inverse of $\operatorname{Shi}(x)=\int_0^x\frac{\sinh(t)}t dt$
Via Fourier Integral:
$\def\Shi{\operatorname{Shi}}$
The hyperbolic sine integral Shi$(x)$ is invertible via complex fourier series converging on $[-L,L]=[-\Shi(l),\Shi(l)]$:
$$f(x)=\Shi(x)\implies f^{...
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$\int_0^\pi t^2\sin(t)\sin(n(\sin(t)-t\cos(t)))dt$ for alternate Goat problem Fourier sine series solution
The goat problem has the following transcendental equation:
$$\sin(a)-a\cos(a)=\frac\pi2,a=1.905695729\dots$$
If $f^{-1}(x)$ is odd, its Fourier sine series of period $\left[-\frac L2,\frac L2\right]$ ...
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Proving $\int_0^{\pi/2-\phi_x} \chi^*\cos\psi\sin\psi\,d\psi = \frac\pi4(1-\sin\phi_x)$, where $\cos\chi^*=\tan \phi_x \tan \psi$
I want to prove the following result
$$
I_1(\phi_x)
=\int_{0}^{\pi/2-\phi_x}
\chi^* \cos{\psi} \sin{\psi} d\psi
= \frac{\pi}{4}[1 - \sin \phi_x ]
$$
where
$\cos \chi^* = \tan \phi_x \tan \psi$.
For $\...
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6
answers
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Evaluation of $~\int_{0}^{2\pi}{\cos(\theta)^2-\sin(\theta)^2\over\sin(\theta)^4+\cos(\theta)^4}\mathrm d\theta$
$$
I:=\int_{0}^{2\pi}{\cos(\theta)^2-\sin(\theta)^2\over\sin(\theta)^4+\cos(\theta)^4}\mathrm d\theta
$$
My tries
$$\begin{align}
s&:=\sin\theta\\
c&:=\cos\theta\\
I&=\int_{0}^{2\pi}{\cos(\...
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Possible to factor this zero-crossing denominator sin(a+b) out of this zero-crossing numerator tan(a-b)?
EDIT: This has been solved.
The reflection coefficient of light approaching glass with an incidence angle $\theta_i$ varies with $$f_R=\frac{\tan(\theta_i-\theta_r)}{\sin(\theta_i+\theta_r)}$$ where $\...
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2
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Find $\int \sin x \cos (\cos x) \cos(\sin x)+\cos x\sin(\cos x)\sin(\sin x)dx$
How do I find: $$\int \sin x \cos (\cos x) \cos(\sin x)+\cos x\sin(\cos x)\sin(\sin x)dx$$? I made this one up by taking the derivative of $-\sin (\cos x)\cos (\sin x)$, and I wonder how someone would ...
- 3,550
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1
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trigonometric substitution in two way
Hi this is one of question that i have got stumbling on.
This is just simple question using trigonometric substitution.
i do understand the solution process.
But What i wanna know is that it uses sin ...
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1
vote
0
answers
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Integrating $\int_0^{\frac{\pi}{2}}\arccos\left(\frac{\cos(x)}{1+2\cos(x)}\right)~\mathrm{d}x$ [duplicate]
Find the integral
$$\int_0^{\frac{\pi}{2}}\arccos\left(\frac{\cos(x)}{1+2\cos(x)}\right)~\mathrm{d}x.$$
Been struggling on this one for ages.
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0
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Proving a Bessel function property using trigonometrical integrals
I need to prove, knowing the (first order) Bessel function:
$$J_n(x) = \frac{1}{\pi}\int_0^\pi \cos(nt - x\sin(t)) \ dt$$
that the following equality is true:
$$J_{n-1}(x) + J_{n+1}(x) = \frac{2n}{x}...
user1107523
1
vote
1
answer
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Calculus 2 Practice Final Bonus Question (definite integral)
Calculate the definite integral $$\int_0^\pi \sin t · \sin^{11}(\cos t) dt$$
It seems straightforward enough yet seems to be a trick question? Using u-substitution, I took $u=\cos t$ and got $\int_1^{-...
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2
votes
3
answers
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Integrate expression with 3 multiplications $\int_0^{\frac\pi4}{x\cos(x)\sin(2nx)}dx$
Specifically I want to integrate the following:
$$\int_0^{\frac\pi4}{x\cos(x)\sin(2nx)}$$
I do know that
$$\frac4{\pi}\int_0^{\frac\pi4}{x\cos(x)\sin(2nx)} = \dfrac{-16\cos(n\pi)}{(4n^2-1)^2{\pi}}$$
I ...
0
votes
1
answer
79
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How to easily visualise $ \int_{-\pi}^{\pi} \cos (nx)\cos(kx)dx=0$?
Is there an easy way to visualise that the following holds?
$$ \int_{-\pi}^{\pi} \cos (nx)\cos(kx)dx=0$$
I thought about simply switching the cosine to sine:
$$ \int_{-\pi}^{\pi} \cos (nx)\sin\left( ...
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votes
0
answers
45
views
Cumulative sum of trigonometric functions
I have two known time series $ Y, 𝜭 $
$Y = \left\{ Y_i \right\}_N \hspace {0.2 in} Y_i > 0 \hspace {0.2 in} -1 \leq Y_{i+1} - Y_i \leq 1 $ for all $i$
$𝜭 = \left\{ 𝜭_i \right\}_N $
Let ...
- 51
3
votes
1
answer
50
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Is this u-substitution valid?
It is not necessary to take any example of any integral, so I'll just drop it: $$u=\sin x; \ \ \ du=\cos x dx$$ which my question is rather here:
Is it possible to do: (squaring both sides) $$u^2=\sin^...
- 29
3
votes
3
answers
111
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Question inside $\int \tan^2 x dx$
$$\int \tan^2 x dx$$ the question isn't behind on how to do it but rather on its solution, let me go through my solution quickly:
If we do a substitution $u=\tan(x)$ then $du=\sec^2 x dx=(u^2+1)dx$ ...
- 29
2
votes
0
answers
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Spectrum of the integral operator $A(f)(x)=\int_{[0,2\pi]}\frac{\sin(n\frac{x-y}{2})}{\sin(\frac{x-y}{2})}f(y)dy$ where $A:L^2(0,2\pi)\to L^2(0,2\pi)$
I want to understand the spectrum of the integral operator $A$ from $L^2(0,2\pi)$ to itself, given by $$A(f)(x):=\int_{[0,2\pi]}\frac{\sin(n\frac{x-y}{2})}{\sin(\frac{x-y}{2})}f(y)dy$$ where $n$ is a ...
- 103