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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How to evaluate the following exponential-trigonometric Integral?
How to evaluate the following Integral? $$I=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{e^{\cos2x}\sin (x+\sin2x)}{\sin x} \, dx$$
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Integral of exponential function multiplied by sine function [duplicate]
Question: Find $\displaystyle\int e^x\sin x\,dx$
Let $I = \displaystyle\int e^x\sin x\,dx$
Then
$$
I = e^x (-\cos x) - \int\bigl[e^x(-\cos x)\bigr]\,dx
$$
Integrating $e^x(-\cos x)$:
$$
I = -e^x\cos ...
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Computing a limit for variable $a\in\mathbb{R}$ using measure theory
So I have this exercise and solutions to it, and lots of questions in bold:
Exercise
Compute the limit
$$
\lim_{n \to \infty} \int_{a}^{+\infty} \frac{n}{1 + n^2 x^2} \, dx
$$
for every $a \in \mathbb{...
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Integrate $\frac1{\sqrt{u^2+v^2} \left(1+u^2+v^2\right)}$ over $[-1,1]^2$?
Problem
$$ \text{Evaluate} \quad I = \iint\limits_{[-1,1]^2} \frac{du\,dv}{\sqrt{u^2+v^2} \left(1+u^2+v^2\right)}$$
The integral is a special case of one that appeared in this recent post, in which ...
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Is there a nice closed form for the integral of $(\tan x)^{2n}$?
I just want to know if there is a nice closed form for this integral:
$$\int_{}^{}(\tan x)^{2n}dx$$
I know that a reduction formula exists and this is what I get by following it:
$$\begin{align*}
\int\...
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Puzzled by asymmetry of cosine integral
I used Mathematica to calculate the antiderivative of $\cos (\pi x)/x$. I obtained the cosine integral
$$
\int \frac {\cos (\pi x)}{x} dx = Ci(x)
$$
where
$$
\begin{aligned}
Ci(x) &:= - \int_x^\...
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Bessel-like integral, with exponential and trigonometric functions involved
Does anyone know if the following integral can be written in a closed form expression and how? Thank you all;
$$I = \int_{\phi=0}^{2\pi} \sin\phi e^{\alpha\cos\phi+\beta\sin\phi} d\phi, \alpha, \beta \...
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The correct formula for the cosine integral of $(1/x^n)cos^m(ax)$
I am not a math major, just a math and physics hobbyist so pardon me if this sounds easy.
I cannot seem to find an exact source to get the answer, even when I asked perplexity, an AI search engine and ...
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Integrating a large product of sines
Recently, I came across the following integral: $\int_{0}^{2\pi}\sin(x)\sin(2x)\sin(3x)\sin(4x)~\mathrm dx=\frac{\pi}{4}$, which can be easily solved by some trigonometry.
But when trying to find a ...
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Integral with singularities
Let $n > 1$ and $1 \le k \le (n-2)$ be integers, and set
$$f(x) := (-1)^n k \left(\frac{k u \sin (k \pi u)}{n-k u}+\frac{(k+1) u \cos \left((k+1) \pi \sqrt{u}\right)}{(k+1)u-n}+\frac{n}{\pi }\...
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Generalized sine integral - How to simplify or solve? [duplicate]
I am new to sine and cosine integrals and I am just doing this out of hobby interest, so please forgive me if this sounds obvious or easy.
I am trying to simplify or solve this expression:
$$\int\frac{...
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Help in solving indefinite integral involving trigonometric functions
Although I found plenty integrals which look similar, e.g. I am struggling with the following indefinite integral:
\begin{equation}
I = \int\frac{1}{(x+cos(x))^{2} + sin^{2}(x)}\,\mathrm{d}x
\end{...
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Evaluating $\int_{0}^{2\pi}\cos ^2(x)\sin(x) \ dx$
How can I find the definite integral of this trigonometric function?
$$\int_{0}^{2\pi}\cos ^2(x)\sin(x) \ dx$$
Currently my thought process is this:
$$ u = \cos x $$
$$du = -\sin x \ dx $$
$$\int_{0}^{...
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Evaluate: $\int{\frac{\cos^4x}{\sin^3x}dx}$ [duplicate]
Evaluate: $$\int{\frac{\cos^4x}{\sin^3x}\ \mathrm dx}$$
My attempt:-
Put $\cos x=t$ so, $-\sin x\ \mathrm dx=\mathrm dt$ ,
Hence,
$I= -{\int{\frac{t^4}{(1-t^2)^2}}dt}$ $=-\int{\frac{t^4-1}{(t^2-1)^2}...
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Evaluate the definite trigonometric integral $\int_{0}^{2\pi} \frac{\sin^2(x)}{1+b\cos(x)}, $ $b \in (0,1)$
$$\int_{0}^{2\pi} \frac{\sin^2(x)}{1+b\cos(x)}, \qquad b \in (0,1)$$
Firstly, I defined $$\tilde{f}(z)=z^{-1}\frac{(\frac{z-z^{-1}}{2i})^2}{1+b\frac{z+z^{-1}}{2}}=\frac{2z^{-1}}{-4}\frac{(z-z^{-1})^2}...
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