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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On the Fourier series expansion of $\sin(\pi x)$ periodic on $(-\frac{1}{2}, -\frac{1}{2})$

For an Undergrad. sophomore Math Methods class I am taking this session, we have recently covered Fourier series expansions. I made good progress in one of the exercises, but I was stuck for days on ...
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Evaluation of $\int\sqrt{1+2\sin(\theta)^2\cos(\theta)^2}\mathrm{d}\theta~~\text{where}~~\theta\in\left[0,{\pi\over4}\right]$

$$\begin{align} I&:=\int \sqrt{1+2\sin(\theta)^2\cos(\theta)^2} \mathrm{d} \theta ~~~\text{where}~~~~\theta\in\left[0,{\pi\over4}\right]\\ \cos(\theta)^2&=1-\sin(\theta)^2\\ I&=\int\sqrt{1+...
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Trigonometric Polynomials and Fourier coefficients.

Can somebody show me how can I approach this tough exercise about Fourier coefficients and trigonometric polynomials? Consider the functions $f(x) =e^{i \pi x/2}$ and $g(x) = \{x\}(1 - \{x\})$ for $x ...
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solve for the unknown in each triangle given and round it to the nearest tenth using the law of simes [closed]

this is the triangle one and two. find the missing or unknown in each triangle so that the triangle will be completed
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Compute $\displaystyle\int_0^{\pi}\dfrac{{\rm d}{\theta}} {\left({\epsilon}\cos\left({\theta}\right)+1\right)^2}$ [closed]

$$\int_0^{\pi}\dfrac{{\rm d}{\theta}} {\left({\epsilon}\cos\left({\theta}\right)+1\right)^2}$$
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What are the steps to get that value integrating the given function?

How to calculate this integral $W=\int_0^{2\pi}\dfrac{6{\epsilon}{\mu}{\omega}{(R/C)^2}\cdot\left({\epsilon}\cos\left({\theta}\right)+2\right)\sin\left({\theta}\right)}{\left({\epsilon}^2+2\right)\...
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what are the steps to get these value integration these functions?

I am working on The Reynolds equation in the case of a long journal bearing with different conditions, and I got stuck on these mathematical calculus, so as seen in the images I know the value of Ws ...
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Why is $\int_{0}^{2\pi} \int_0^{2\pi} \frac{\ln(21-4(\cos x+\cos y+\cos(x+y)))}{2\ln(9/2)}\frac{dx}{2\pi} \frac{dy}{2\pi}$ almost $1$?

Consider the function $$ f(x,y) = \frac{\ln(21-4(\cos(x)+\cos(y)+\cos(x+y)))}{2\ln(9/2)} $$ Its average value is awfully close to unity: $$ \int_{0}^{2\pi} \int_0^{2\pi} f(x,y) \frac{\mathrm dx}{2\pi} ...
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Definite Integral of $\int_{-L}^{L} \cos(\frac{nπx}L)\cos(\frac{mπx}L)dx$

Please can someone explain why I need to substitute the values of integers $m$ and $n$ in the definite integral: $$I=\int_{-L}^{L} \cos\left(\frac{nπx}L\right)\cos\left(\frac{mπx}L\right)dx = \begin{...
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Columbia Integration Bee 2022 Finals [duplicate]

I want to find the definite integral shown below, but I'm not quite sure where to start. The fastest solution apparently involved some sort of change of variables, but I can't quite find a ...
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Trigonometric integral without using tangent substitution

How can we solve integrals of rational functions of trigonometric functions like $$\int \frac{1}{3 \sin{x} + 5\cos{x}}dx$$ without tangent (Weierstrass) substitution? I assume that the polynomials are ...
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Solve the integral $\int_{-\frac{\pi}2}^{\frac{\pi}2} \frac{\sin^3x}{\tan^3x+\cot^3x} dx$

Question Solve the integral,$$\int_{-\frac{\pi}2}^{\frac{\pi}2} \frac{\sin^3x}{\tan^3x+\cot^3x} dx$$ Attempt I converted the equation in terms of $\sin(x)$ and $\cos(x)$ using the definition of $\tan(...
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Is this a valid method in Trig Substitution to skip steps?

I've been practicing for a test tomorrow, and with the past few Trig Sub questions I have done involving $\sqrt{x^2 + a^2}$, $sec(u)$ always equals the square root in the expression. Is this true for ...
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Evaluating $\int\frac{3+\cos(x)}{2-\cos(x)}dx$

I am trying to find $$\int\frac{3+\cos(x)}{2-\cos(x)}dx$$ I did long division and got the integrand to be $$-(1+\frac{5}{\cos(x)-2})$$ To simplify the second term, I tried conjugate multiplication as ...
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Asymptotic expansion of cosine/sine integral

For $a,b$ in some compact interval $I$, consider the integral function $K_{a,b}:\mathbb{R}\rightarrow\mathbb{R},$ defined by $$K_{a,b}(x)=\displaystyle\int\limits_{0}^{\pi} \cos(ax\cos(t))\sin(bx\sin(...
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Integrals, and even and odd functions

I am looking at a proof and got stuck on a part with an integral. I tried to simplify the problem as much as possible, I hope I did not omit any potential helpful information. I have an even ...
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Trigonometric integral inequality

Let us consider the integral equation \begin{equation} f(x)=\lambda \int_{0}^{\pi} \cos (x-y) f(y) \hspace{1mm}dy+g(x), \quad x \in[0, \pi], \end{equation} where $f$ is an unknown function on $[0, ...
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I have the integral $\int_{-1}^{1} \frac{\arccos(x)}{1+x^2} \,dx $ and some questions. Any help appreciated!

$$\int_{-1}^{1} \frac{\arccos(x)}{1+x^2} \,dx $$ Hi everyone! Sorry for my poor formatting skills, I'm still quite new to this platform. I do not know how to solve this integral. Things that I tried ...
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Evaluate $\int_0^{\pi/2} \sqrt{ \cos x}\>\cos\frac{3x}2 \>\cos(2n+1)x\> dx$

I am interested in the trigonometric integral below $$I_n = \int_0^{\pi/2} \sqrt{ \cos x}\>\cos\frac{3x}2 \>\cos(2n+1)x\> dx $$ A direct integration appears a bit too much. With the help of ...
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What is the integral $\int_0^{π/2} \frac{\cos^2(x)}{\sin(x)+\cos(x)}dx$: [closed]

I've been trying to figure out this integral and I'm completely stuck, I'd be really grateful if anyone could help out! It's from a 2020 integration bee. Edit: thanks for the answer, it was really ...
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Integral of the $n$-th root of $\tan(x)$

Recently, I’ve started doing calculus and have ventured my way into the tedious integration of $\tan(x)$ to some specific root. I’ve started wondering whether there is any generalisation of the ...
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How to do $ \int -x \sec(2x) dx $?

I know how to solve integration by parts of some functions, but I came across the integration of $-x\sec(2x)$, and I can't solve it. I put the question on an online calculator. I got the answer, but I ...
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Solving $ \int \sqrt{4\sin^2(x) + \cos^2(x) } dx $ without an elliptical integral

Solving: $$ \int \sqrt{4\sin^2(x) + \cos^2(x) } dx $$ I have used calculators attempt this problem, but they all use the elliptical integral of the second parameter. My goal is to somehow solve this ...
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Is it possible to express $\sin(\frac{k\pi}{4})$ as something of the form $(-1)^{f(k)}$?

For example $\cos(k\pi)=(-1)^k$ similarily, $\sin(\frac{k\pi}{2})=(-1)^\frac{k-1}{2}$ when k is odd, and 0 otherwise. Is there a similar representation for $\sin(\frac{k\pi}{4})$
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Computing the integral $\int_{-\pi}^{\pi}\frac{1}{2-\cos(t)-\sin(t)}\;dt$

Let $a, b \in \mathbb{C}$, such that $\lvert a \lvert < 1 < \lvert b \lvert$, and $\gamma(t)=e^{it}$, $t\in [0,2\pi]$. Show that $\int_\gamma \frac{1}{(z-a)(z-b)}dz=\frac{2\pi i}{a-b}$ Using 1),...
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Trigonometric integral with square roots

When studying a fixed-point problem for a certain integral equation which arises in my research, I am led to evaluating an integral of the form $\int_0^{\pi} e^{-\sqrt{a^2-\cos t}} \cos t\,dt$ where $...
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What type of trigonometric integral is $\int_{-\pi}^{\pi} \sin^{2}(x)\sin(n\pi x)\,dx$?

None of the trigonometric integrals look like the integral I have written. I need use some type identity trigonometric ? $$ \int_{-\pi}^{\pi} \sin^{2}(x)\sin(n\pi x)\,dx $$
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About a trigonometric integral

I play with Maple and i am interested about the integral $\int_{0}^{\infty }\!{\frac { \left( {x}^{2}+{z}^ {2} \right) }{{{\rm e}^{a\pi\,x}}-1}\sin \left( 2\,\arctan \left( { \frac {x}{z}} \right) \...
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Solving integral $\int \frac{\sqrt{x^2 + x}}{x}dx$ (problem 36 in section $6.25$ in Tom Apostol's calculus)

Integrals which involve $\sqrt{(cx + d)^2 - a^2}$ could often be simplified if we do a substitution $cx + d = a \sec t$. If we take a concrete example, $\int \frac{\sqrt{x^2 + x}}{x}dx$, then the ...
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Integral of $(1+\cos(t))^n$ from $-\pi$ to $\pi$

Using an integration solver (Mathematica), I've gotten the following result $$\int_{-\pi }^{\pi } (1+\cos (t))^n \, {\rm d}t = 2^{1-n} \pi \binom{2 n}{n}$$ Any suggestion on how to prove this would be ...
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Trig sum and residues

I am interested in the sum $$ \sum_{n=1}^k 2\biggl[\sin\biggl(\frac{n\pi}{2k+2}\biggr)\biggr]^{-2g} $$ where $k,g$ are integers. It is not too hard to show that this can also be expressed as $$ -1-2\...
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9 votes
1 answer
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Integrate $\int_0^\infty \frac x{ \sec x\cosh x \>+\>1}dx$

I am interested in whether it is possible to evaluate the integral $$\int_0^\infty \frac x{ \sec x\cosh x +1}dx$$ For reference, the analogous integral below is manageable $$\int_0^\infty \frac 1{ \...
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Integral of a product of cosines with different arguments?

Hello all I'm working through a guided proof of Plancherel's Theorem and am currently trying to determine why the the following integral gives the result it does: $\int_{-a}^{a} cos(\frac{n\pi x}{a})...
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How to prove that $\sum_{ k= 1}^{N-1} \sin(\frac{k}{N} \pi)^{-2} = (N^2 - 1)/3$? [duplicate]

I find the equality \begin{equation} \sum_{k = 1}^{N-1} \sin(\frac{k}{N}\pi)^{-2} = \frac{N^2 - 1}{3}, \end{equation} during study. Just wonder how can I prove it? Short matlab code to do the ...
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Integrate $\int_0^{\infty} \frac{\sin^2 x}{\cosh x\>+\>\cos x}\frac{dx}x $

It is known that (see for example) \begin{align} &\int_0^{\infty} \frac{\sin x}{\cosh x+\cos x}\frac{dx}x =\frac\pi4\\ &\int_0^{\infty} \frac{\sin^3 x}{\cosh x+\cos x}\frac{dx}x =\frac\pi8 \...
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Average value of trigonometric functions

Suppose I have to find out $\langle \sin^2\theta\rangle$ and $\langle\cos^2\theta\rangle$. What I do normally would be : $$\langle\sin^2\theta\rangle=\frac{1}{2\pi}\int_{0}^{2\pi}\sin^2\theta d\theta=\...
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How to show $\int_0^{\infty} \frac{\sin^3 x}{\cosh x\>+\>\cos x}\frac{dx}x=\frac{\pi}{8}$

I would like to evaluate the integral below$$\int_0^{\infty} \frac{\sin^3 x}{\cosh x+\cos x}\frac{dx}x $$ which I found to be $\frac \pi8$ numerically. I was able to evaluate a similarly looking, yet ...
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3 votes
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Discrete sum "wave packet" for $x\to\infty$

Let $A:\mathbb{R}\to\mathbb{R}$ be a continuous function with compact support having a continuous derivative. In particular $A(k)=0$ outside some sufficiently wide interval $[k_0,k_1]$. Define a (...
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Integration of mixed sin and cos functions - can't find it

I can't remember the question that was posted around Feb 2nd,2022 I presume that it has been deleted by some reasons The question was : $$ \int \frac{\sin (\cos x)}{\cos(\sin x)} dx $$ can anyone ...
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1 answer
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How to solve complex triple integrals involving square roots

I came a cross this problem while doing calc homework: Use a triple integral to find the volume of the solid bounded below by the cone $z = \sqrt{x^2 + y^2}$ and bounded above by the sphere $x^2 + y^...
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1 answer
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How do I prove $\int_0^\pi tf(\sin t) dt = 1/2 \int_0^\pi f(\sin t) dt$?

I am not really sure how to approach this. I have tried to use integration by parts but it does not seem to work. Thank you
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How do we obtain the integral of $\tan(x)$: is it $-\ln\left|\cos(x)\right|$ or $\ln\left|\sec(x)\right|$?

Is the integral of $\tan(x)\,\mathrm{d}x$ equal to the negative $\ln$ of absolute value of $\cos(x)$, the same as integral of $\tan(x)\,\mathrm{d}x$ equal to the $\ln$ of absolute value of $\sec(x)$ $$...
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9 votes
2 answers
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Why can't I use trig substitution for this integral?

$ \int \frac x {\sqrt {1-x^2}}dx $ I was attempting to solve this integral, and it would appear the solution to it is $-\sqrt{1 -x^2}+C$. When I attempted to solve it, however, I attempted to let $x = ...
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2 votes
1 answer
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Where does the absolute value come from in this integral evaluation?

I have an integral which can be evaluated by trig substitution: $$\int\frac{1}{x\sqrt{x^2-4}} dx.$$ I let $x=2\sec(u)$ so that we have $$\int\frac{2\sec(u)\tan(u)}{2\sec(u)\sqrt{4\sec^2(u)-4}} du=\int\...
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Axler 6.9: Show that the following list is orthonormal.

Let $n \in \mathbb{Z_{+}}$ and show that the list below is an orthonormal list of vectors in $C[-\pi,\pi]$ in the vector space of real valued functions on $[-\pi,\pi]$ with the inner product given ...
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3 votes
3 answers
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How to find the right answer for Integral of $\sin(2x)\cos(2x)$

This is what I did when I solved the $$\int(\sin(2x)\cos(2x))dx$$ First I used integration and made $$u=2x$$ Then I found the derivative of $2x$ and determined its value when it is equal to $dx$, $$\...
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Prove $ \int_0^1 (1+a+a^2\pi^2x^{2a})\sin(\pi x^a)dx = a\pi$ using previous results

If $a>0$ prove, $$ \int_0^1 \sin(\pi x^a)dx+a\pi\int_0^1 \ x^a\cos(\pi x^a)dx = 0 \tag1$$ $$ \int_0^1 \cos(\pi x^a)dx-a\pi\int_0^1 \ x^a\sin(\pi x^a)dx = -1 \tag2$$ $$ \int_0^1 (1+a+a^2\pi^2x^{2a})...
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-4 votes
1 answer
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$\int_{0}^{\pi} \frac{x \tan x}{\sec x \tan x} d x$

I have solved the Q. Answer in my textbook is $-2\pi + x $ but I got $-1$. So , I want to confirm my answer and understand where I have done mistake. $$\int_{0}^{\pi} \frac{x \tan x}{\sec x \tan x} d ...
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Show that a definite integral is non negative.

Consider the definite integral $$I=AB+CD$$ where $$ A=\int_{1}^{100}\sinh\left(\frac{\ln x}{8}\right)\cos \left(\frac{\ln x}{2}\right)dx $$ $$ B=\int_{1}^{100}\cosh\left(\frac{\ln x}{8}\right)\cos \...
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Contour integration on the interval [0,pi/4]

I know how to apply contour integration to definite trigonometric intervals on the interval on the interval [0,2pi], but I am curious if this works on any other intervals like [0,pi], or [0,pi/4] like ...
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