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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Integral $\int_0^{\frac{\pi}{2}}\frac{\log\left(\sin x\right)}{\cos^2x+y^2\sin^2x}{d}x=-\frac{\pi}{2}\frac{\log\left(1+y\right)}{y}$

Prove that $$\int_0^{\frac{\pi}{2}}\frac{\log\left(\sin x\right)}{\cos^2\left(x\right)+y^2\sin^2\left(x\right)}{\rm d}x=-\frac{\pi}{2}\frac{\log\left(1+y\right)}{y}$$ where $y\ge0$. I came across this ...
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examples of trigonometric functions $f(x)$ such that $\int_0^M e^{if(x)} dx$ becomes a desired value

Let $f(x)$ be a non-constant differentiable real-valued function defined on $x \in [0,M]$. What are some examples of trigonometry functions $f(x)$ such that it satisfies the following equations: \...
user185671631's user avatar
-3 votes
0 answers
37 views

How to differentiate a trig value? [closed]

This is a a function concerning the differentiation of a trig value $\sin^{2}{x}$ Does anyone knows what is its derivative? $g(x)=\sin^{2}((\frac{i}{(1-3x)})^{\frac{1}{2}})$
HaoLiang Quan's user avatar
4 votes
2 answers
205 views

Complex Integral ML Lemma

I must solve $$\int_{-\infty}^{\infty}\frac{x \text{sin}x}{x^2+4} dx$$ I simplified this to $$\int_{-\infty}^{\infty}\frac{x \text{sin}x}{x^2+4} dx = \frac{1}{i}\int_{-\infty}^{\infty}\frac{x e^{ix}}{...
adisnjo's user avatar
  • 195
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1 answer
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Why is the result of calculating definite integrals in this way incorrect

let $$t=\cos x$$ then $$ \int_{-1}^1{\mathrm{arc}\sin \sqrt{1-t^2}}\mathrm{d}t=\int_0^{\pi}{x\sin x}\mathrm{d}x=\pi $$ Why is it wrong? The correct result is $$ \int_{-1}^1{\mathrm{arc}\sin \sqrt{1-t^...
yu song's user avatar
3 votes
2 answers
158 views

How to reduce $\int_0^{\pi/2}\frac{1-\sin x}{\cos^2x}\sqrt{\tan x}\,dx$ to complete elliptic integral?

I came across another old post concerning a definite integral whose closed form can be expressed with a complete elliptic integral: $$I = \int_0^\infty \left(\sqrt{1+x^4} - x^2\right) \, dx = \frac1{6\...
user170231's user avatar
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1 vote
1 answer
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Calculating the integral $\int_{0}^{2\pi} \frac{\cos^2(3\theta)}{5-4\sin(2\theta)}d\theta$ using residue theorem [duplicate]

I'm trying to solve this integral: $$ \int_{0}^{2\pi} \frac{\cos^2(3\theta)}{5-4\sin(2\theta)}d\theta $$ [UPDATE: Note this is not the same integral as the one solved here even though very similar]. ...
giorgio's user avatar
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2 votes
0 answers
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Alternative approach for $\int\frac{1}{\tan\left(\frac{1}{2}\csc^{-1} (x)\right) - \tan\left(\frac{1}{2}\sec^{-1}(x)\right)} \, dx$

This is example 2 in my "Integration Using Some Euler-Like Identities" blog post. $$\int\frac{1}{\tan\left(\frac{1}{2}\csc^{-1} (x)\right) - \tan\left(\frac{1}{2}\sec^{-1}(x)\right)} \, dx\...
Emmanuel José García's user avatar
12 votes
1 answer
571 views

Prove $\int_0^\pi\arcsin(\frac14\sqrt{8-2\sqrt{10-2\sqrt{17-8\cos x}}})dx=\frac{\pi^2}{5}$.

There is numerical evidence that $$\int_0^\pi\arcsin\left(\frac14\sqrt{8-2\sqrt{10-2\sqrt{17-8\cos x}}}\right)dx=\frac{\pi^2}{5}.$$ How can this be proved? Context In another question, three random ...
Dan's user avatar
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1 answer
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$\lim_{k\to\infty}\frac{1}{k}(\sum_{i=1}^k \sin(\omega i)\sin(\omega(i-j))=\frac{1}{2\pi} \int_{0}^{2 \pi}\sin (\omega x)\sin(\omega(x-j))dx, j\ge0$ [closed]

Is this a standard result with a well known proof? Or would anyone know how to go about proving it? $$\lim_{k\to\infty}\frac{1}{k}(\sum_{i=1}^k \sin(\omega i)\sin(\omega(i-j))=\frac{1}{2\pi} \int_{0}^{...
jjjjjames's user avatar
2 votes
1 answer
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How to integrate $\int_{-\infty}^{\infty} e^{irx}\frac{1-\cos x}{\pi x^2} dx$

Note: This is for an assignment. I am not asking for solutions, just for ideas of where to look because everything I've tried has been a dead end. I have to evaluate the integral: $$\int_{-\infty}^{\...
stephan's user avatar
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19 votes
1 answer
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Prove $\int_0^1\frac{1}{\sqrt{1-x^2}}\arccos\left(\frac{3x^3-3x+4x^2\sqrt{2-x^2}}{5x^2-1}\right)\mathrm dx=\frac{3\pi^2}{8}-2\pi\arctan\frac12$.

There is numerical evidence that $$I=\int_0^1\frac{1}{\sqrt{1-x^2}}\arccos\left(\frac{3x^3-3x+4x^2\sqrt{2-x^2}}{5x^2-1}\right)\mathrm dx=\frac{3\pi^2}{8}-2\pi\arctan\frac12.$$ How can this be proved? ...
Dan's user avatar
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1 answer
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Solving $\int_0^{\pi} \sin^2(x) \sin(nx) \sin(mx) dx$ for general $n, m$.

It is well known that $\int_0^\pi \sin(nx) \sin(m x)dx = \dfrac{\pi}{2}I(m = n)$. By using an online integration calculator, I found that $$\int_0^\pi \sin^2(x) \sin(nx) \sin(mx) = \dfrac{\pi}{4}I(m=n)...
John Choi's user avatar
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3 answers
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Can we reduce $\int_0^{\pi/2}\frac{\sqrt{\sin x}}{1+\cos x}\,dx$ to complete elliptic integrals?

This definite integral has an equivalent closed form in terms of complete elliptic integrals, $$\begin{align*} I &= \int_0^\tfrac\pi2 \frac{\sqrt{\sin x}}{1+\cos x} \, dx \\ & = 2 - \sqrt{\...
user170231's user avatar
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1 vote
1 answer
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Integral of rational trigonometric function

I would need some help trying to evaluate the following integral: $$ I = \frac{1}{\pi}\int_0^{\pi} \frac{1-\cos \left(2n u\right)}{2 \cos \left(u\right)-x}\mathrm{d}u, $$ where $|x|>2$ and $n=1,...
Matt's user avatar
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2 answers
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How to compute integral $\int_0^\pi \ln\left(2\sin\frac x2\right)dx$ [closed]

I would like to compute $$\int_0^\pi \ln\left(2\sin\frac x2\right)dx$$ The primitive of the integrand does not seem to have a closed form. So I guess one would need to use symettry or trig identities ...
user10000024's user avatar
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Chebyshev polynomials orthogonal with respect to different weight function?

The following exercise appears in Ridgway Scott's Numerical Analysis: Where $\omega_n(x)$ is the Chebyshev Polynomial of the first kind, that is $$\omega_{n+1}(x)=2^{-n}\cos((n+1)\cos^{-1}(x)$$ I ...
modz's user avatar
  • 101
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1 answer
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Question on manipulation step of trigonometric integral

In our book we have the manipulation \begin{align*} b\int\limits_0^{2\pi}\sqrt{\left(1-\left(1-\frac{a^2}{b^2}\right)\sin^2(t)\right)}~\!dt=4b\int\limits_0^{\frac{\pi}{2}}\sqrt{\left(1-\left(1-\frac{a^...
Philipp's user avatar
  • 4,493
2 votes
4 answers
188 views

Show that $\int_0^{\frac\pi 2}\frac{\theta-\cos\theta\sin\theta}{2\sin^3\theta}d\theta=\frac{2C+1}4$

While trying to evaluate $\int_0^1 k^2K(k)dk$ related to elliptic integral of the first kind, by integral switching method, I reached the trigonometric integral $$\int_0^{\frac\pi 2}\frac{\theta-\cos\...
Bob Dobbs's user avatar
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Evaluating $\int_1^2 \frac{1}{\tan{\left(\frac{\sec^{-1}(x)}{2}\right)}-x}\,dx$

In a previous question of mine (see here and here), I suggested the following general transformation for integrals involving $\tan{\left(\frac12\csc^{-1}{(x)}\right)}$ or $\tan{\left(\frac12\sec^{-1}{(...
Emmanuel José García's user avatar
6 votes
2 answers
236 views

Tricky integral $\int_0^\pi{12\cos x\ \mathrm{sech}(\frac \pi2 \tan\frac x2)}\mathrm{d}x=\pi^2$

I need to show that the following tricky integral: $$\int_0^\pi{12\cos x\ \mathrm{sech}\left(\frac {\pi}2\tan\frac x2\right)}\mathrm{d}x$$ is equal to exactly $\pi^2$. I have no idea how to start. I ...
Amit Zach's user avatar
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4 votes
1 answer
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Tricky integral involving Arctan

I'm trying to do an integral which arises from another integral that is simple in polar coordinates (so I can work out the answer that way). However I am using cartesian coordinates and trying to see ...
su.jai's user avatar
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10 votes
2 answers
374 views

Show that $\int_0^\pi\arctan\left(\frac{a\cos x+b\sin x}{c\cos x+d\sin x}\right)dx=\pi\arctan\left(\frac{ac+bd}{|bc-ad|+c^2+d^2}\right)$

Given that $a,b,c,d\in\mathbb{R}$ and that $c$ and $d$ are not both $0$, show that $$\int_0^\pi\arctan\left(\frac{a\cos x+b\sin x}{c\cos x+d\sin x}\right)\mathrm dx=\pi\arctan\left(\frac{ac+bd}{|bc-ad|...
Dan's user avatar
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2 votes
3 answers
177 views

Alternative method of evaluating $\int_0^{\frac{\pi}{2}} \sin ^{2 n} x \ln (\tan x) d x $?

LATEST EDITION Glad to share with you that we had found below as an answer, in general, that $$\boxed{ \int_0^{\frac\pi2} {\sin^n x} \ln{(\tan x)} \,dx =\frac{\sqrt{\pi}}{4 \Gamma\left(\frac{n}{2}+1\...
Lai's user avatar
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0 votes
1 answer
35 views

$\int \frac{\sqrt{a^2-x^2}}{x^2}dx$ using trigonometric substitition

I'm aware this question is more easily done using substitution by parts or euler substitution, but this was under a section in my book where we were asked to use trigonemtric substations. Substituting ...
Kryptic Coconut's user avatar
0 votes
2 answers
69 views

Ratio of definite integrals

my school gave this question out and I couldn't think of a way to do it other than bashing by parts multiple times. Is there a clever solution to the question? Given: $$ I = \int_{0}^{4\pi} e^x (\sin^...
Kevin's user avatar
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3 votes
3 answers
173 views

Is there an easier way to evaluate the integral $I=\int_{0}^{1}\frac{\ln x\sin^{-1}x\cos^{-1}x}{x}dx$?

As I was surfing the Mathematics side of Instagram (as usual), I came across this integral: $$I=\int_{0}^{1}\frac{\ln x\sin^{-1}x\cos^{-1}x}{x}dx$$ It encouraged me to embark on a very satisfying ...
Kisaragi Ayami's user avatar
1 vote
1 answer
36 views

How to Prove the Divergence of an Improper Integral Involving Absolute Value

I'm working on understanding the convergence properties of certain improper integrals and encountered the following integral: $$\int_{0}^{\infty} \left| \frac{\cos(x)}{\sqrt{x}} \right| \, dx$$ I ...
Matan Bitton's user avatar
3 votes
1 answer
82 views

Solve trig integral $\int_{0}^{\pi/2} \left(\frac{\sin5x}{\sin x}\right)^2 \,dx $ [duplicate]

I've stumbled across this integral: $$\int_{0}^{\pi/2} \left(\frac{\sin5x}{\sin x}\right)^2 \,dx $$ I was on a time limit and my intuition told me: $$\int_{0}^{\pi/2} \left(\frac{\sin5x}{\sin x}\right)...
Avgustine's user avatar
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3 votes
2 answers
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How to calculate: $ \int_{-7 \pi}^{2 \pi} \frac{1}{2 \sin(x) - \cos(x) + 5} dx$

How to calculate the value of this Riemann integral? $$ \int_{-7 \pi}^{2 \pi} \frac{1}{2 \sin(x) - \cos(x) + 5} dx $$ I used the universal trigonometric substitution and found the following ...
Jacobs Monarch's user avatar
0 votes
1 answer
33 views

Why is $2\sin\theta$ the upper limit for this inner integral, and why is $0$ the lower limit?

I am having difficulty understanding how this practice question is supposed to be solved. Here it is: Use polar coordinates to find the exact value of the double integral, $\iint_D x$ $dA$ where $D$ ...
Lim Min Kang's user avatar
2 votes
1 answer
71 views

Is $\int_0^{\frac\pi 2}\frac{dx}{1-k^2\sin^2x}$ an elliptic integral?

Let $\Delta=\sqrt{1-k^2\sin^2x}$, $E(k)=\int_0^{\frac\pi 2}\Delta dx$ and $K(k)=\int\frac{dx}{\Delta}.$ Start wearing purple, gives a nice answer for the interesting identity $\int_0^{\frac\pi 2}\frac{...
Bob Dobbs's user avatar
  • 11k
7 votes
1 answer
298 views

Evaluating $\int_0^{\pi } \frac{\sin (n \sigma )}{(a-\cos \sigma )^2} \, d\sigma$

What is the formula for $$\int_0^{\pi } \frac{\sin (n \sigma )}{(a-\cos \sigma )^2} \, d\sigma$$ where $ a>1 $ and $n$ is a positive integer? To evaluate, I tried to replace $a$ with $\cosh\xi $ in ...
Rajai's user avatar
  • 81
7 votes
3 answers
168 views

A curious result from Mathematica on $\int\sin^5x\cos^7x \,dx$

While creating an answer key for my Calc 2 students and trying to save a bit of time, I plugged a trigonometric integral into Mathematica and was a little confused about how it approached the problem. ...
jgd1729's user avatar
  • 434
2 votes
3 answers
159 views

Show that $\int_{0}^{\infty}\sin\left(\frac{1}{x^{2}}\right)\ln xdx=\sqrt{\frac{\pi}{2}}\left(\frac{\gamma}{2}+\frac{\pi}{4}+\ln2-1\right)$

I came up with this problem while messing around in Desmos and was very surprised by the solution I got from Wolfram Alpha. The integral calculator is unable to solve it and WolframAlpha does not ...
Dylan Levine's user avatar
  • 1,594
0 votes
0 answers
67 views

Trigonometric substitution with $\sec{\theta}$

My question is about the integration technique - trigonometric substitution $x = \sec{\theta},$ with $x \in (0, \frac{\pi}{2}) \cup (\frac{\pi}{2},\pi)$, in accordance with how the inverse secant ...
Joseph's user avatar
  • 560
4 votes
1 answer
67 views

A trigonometric integral and algebraic integral limit of the same kind

Evaluate the value of: $$(i)\lim_{n \to ∞} n^2 \left(\int_0^{\pi/2} (\sin^n(x)+\cos^n(x))^{1/n} dx-\sqrt2 \right)$$ $$(ii)\lim_{n \to ∞} n^2 \left(\int_0^1 (x^n+(1-x)^n)^{1/n} dx-3/4 \right)$$ Both of ...
Cognoscenti's user avatar
3 votes
1 answer
122 views

Show that $\int_0^{\frac{\pi}2}K(\sin t)dt=\frac{\Gamma(\tfrac14)^4}{16\pi}$

By using the Maclaurin series $K(k)==\frac{\pi}2\sum_{n=0}^\infty c_n ^2 k^{2n}$ where $c_{n}={2n\choose n}2^{-2n}$ we have $$\int_0^{\frac{\pi}2}K(\sin t)dt\\ =\frac{\pi}2\sum_{n=0}^\infty c_n ^2\...
Bob Dobbs's user avatar
  • 11k
1 vote
0 answers
83 views

Ask for solutions to two improper integrals whose integrands contain the integer powers of the sine function

Let $k\in\mathbb{N}$, $\alpha\in\mathbb{R}$, and $a,b\in\mathbb{R}$ such that $a,b>0$. Then \begin{equation}\label{alpha-n=1-k-odd}\tag{q1} \int_0^\infty\frac{\sin^{2k-1}t}{t}\sin(\alpha t)\...
qifeng618's user avatar
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3 votes
1 answer
170 views

How to integrate $\int_0^{\pi/2} \frac{x(1+\sin^2 x)\cos x}{(3+\sin^2 x)(1+3\sin^2 x)}\,dx$

Question How to integrate $$\int_0^{\pi/2} \frac{x(1+\sin^2 x)\cos x}{(3+\sin^2 x)(1+3\sin^2 x)}\,dx$$ My attempt \begin{align*}I &= \int_0^{\pi/2} \frac{x(1+\sin^2 x)\cos x}{(3+\sin^2 x)(1+3\sin^...
Sbsty 's user avatar
1 vote
2 answers
44 views

Problem on trigonometric substitution of an integral

Suppose we have to find the integral of the function $\sqrt{\frac{x}{1-x}}$. To solve this, I set $\sqrt{x} = -\sin\theta$, where theta belongs to [0, -π/2) which implies that $x = \sin^2\theta$. The ...
Rajesh Paul's user avatar
2 votes
1 answer
89 views

What is $\int_{0}^{\infty}\left(\frac{\pi}{2}-\arctan\left(x^{a}\right)\right)dx$?

I came up with this myself while messing around on Desmos and an Integral Calculator. The solutions for $I_a=\int_{0}^{\infty}\left(\frac{\pi}{2}-\arctan\left(x^{a}\right)\right)dx$ get crazy kind of ...
Dylan Levine's user avatar
  • 1,594
18 votes
5 answers
4k views

Why am I getting a finite integral for infinite area?

The following is the graph of the $\tan(x)$ function: We can clearly see there how it’s undefined at $\frac \pi 2$. Now, what if we wanted to find the area between the tangent function and the x-axis ...
user avatar
3 votes
2 answers
166 views

Integrals of the form $\int_0^1\text{arctanh}^a(y) y^{2b} dy$

Recently I have been trying to calculate integrals of the form: $$ I(a,b)=\int_0^1\text{arctanh}^a(y) y^{2b} dy$$ for some positive integer-valued $a$ and $b$. The values $a=0$ or 1 have quite trivial ...
Fred Li's user avatar
  • 642
5 votes
1 answer
232 views

Evaluate $\int_{-\infty}^{\infty}\sin^{-1}\left(\frac{\sin(x)}{x}\right)dx$

I was curious about this integral so I asked about its convergence here: Does $\int_{-\infty}^{\infty}\sin^{-1}\left(\frac{\sin(x)}{x}\right)dx$ converge? @jizert answered, explaining that it did ...
Dylan Levine's user avatar
  • 1,594
2 votes
2 answers
213 views

Does $\int_{-\infty}^{\infty}\sin^{-1}\left(\frac{\sin(x)}{x}\right)dx$ converge?

My attempts: $$\frac{dy}{dx}=\sin^{-1}\left(\frac{\sin(x)}{x}\right)$$ $$\sin\left(\frac{dy}{dx}\right)=\frac{\sin(x)}{x}$$ $$x\sin\left(\frac{dy}{dx}\right)=\sin(x)$$ Seems like a dead end. $$\int_{-\...
Dylan Levine's user avatar
  • 1,594
3 votes
1 answer
582 views

Evaluating $\int_{2}^{5}\sqrt{\tan{\left(\frac{\csc^{-1}(x)}{2}\right)}}\,dx$ using a new (?) trick

The following identities have been suggested based on formulas in a previous question of mine. If complex $\theta_1=\cos^{-1}(p)$ and $\theta_2=\sec^{-1}(p)$, where $p\in(-1, 0) \cup (1, \infty)$, ...
Emmanuel José García's user avatar
1 vote
1 answer
120 views

How to generalize this problem for powers of sine function?

How generalize the following? Evaluating two improper integrals involving powers on $\sin(t)$ How to generalize the above for higher powers of sine(different powers for the sine function, since ...
dragonofkia's user avatar
3 votes
1 answer
170 views

Show $\int_{0}^{\infty}\frac{\sin x^{2}}{\pi-x^{2}}dx=\frac{\sqrt{\pi}}{2}\left[C(\sqrt{2})+S(\sqrt{2})\right]$

Show that: $$\int_{0}^{\infty}\frac{\sin x^{2}}{\pi-x^{2}}dx=\frac{\sqrt{\pi}}{2}\left[C(\sqrt{2})+S(\sqrt{2})\right]$$ where $C$ and $S$are the Fresnel integrals defined as: $$\displaystyle C(u)=\...
Sbsty 's user avatar
1 vote
1 answer
119 views

How to solve this integral trigonometric integral or prove that is does not have a closed form solution? [closed]

How would I solve this. $$I= \int_0^{\frac{\pi}{4}} \sqrt{\sec^2(x)+\sin^2(x)}\ dx$$ The approximate value of the integral: $$I = 0.937723050585$$ I am trying to solve this integral to find the arc ...
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