Questions tagged [integration]

Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of the integrals being considered. This tag often goes along with the (calculus) tag.

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A question about the derivation of Poisson formula with respect to 2-dimensional wave equation

In section 2.4 (wave equation) of Partial Differential Equation (Evans), the author used the descent method to derive the Poisson's formula for two-dimensional wave equation. For the $n=3$ case, the ...
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What does $d^n\textbf{x}$ mean in this context?

I found the following on Wikipedia. Integration over more general domains is possible. The integral of a function $f$, with respect to volume, over an $n$-dimensional region $D$ of $\mathbb{R}^{n}$ ...
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1 answer
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Showing $\int _{0} ^{\pi/4} \frac{\cos^{2022}(x)}{\sin^{2022}(x) + \cos^{2022}(x) } dx \approx \frac{\pi}{4}$

Show that $$\int_{0} ^{\pi/4} \frac{\cos^{2022}(x)}{\sin^{2022}(x) + \cos^{2022}(x) } dx \approx \frac{\pi}{4}$$ My method was this: I tried using $x \to \pi/4-x$ conversion but that doesn't lead to ...
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3 votes
1 answer
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Evaluate the algebraic indefinite integral $\int\frac{\sqrt{22+10x+x^2}}{x}dx$

$$\int \frac{\sqrt{22+10x+x^2}}{x}dx$$ This is what I tried $$\int\frac{\sqrt{(x+5)^2-3}}{x}dx$$ I substituted $x+5=\sqrt{3}\sec\theta$ then got this $$\int\frac{3\tan^2\theta\sec\theta}{\sqrt{3}\sec\...
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1 vote
1 answer
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Using Riemann sums to approximate the second antiderivative

I’m currently working on a coding project where I’m given the the net force acting on an object at any time $t$ (meaning I essentially have its acceleration). I know the object’s current position and ...
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Calculate the integral where $P_{n}$ and $P_{m}$ are Legendre Polynomials

Calculate the folowing integral: $$I_{k,m}=\int_{-1}^{1} x(1-x^2)P'_{n}(x)P'_{m} dx $$ So, my attempt to solve this consisted in: First, I thought of manipulating the folowing relations so i could get ...
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Where am I going wrong integrating $\int x^2(x^3-6)^{34} dx$ by parts?

EDIT : Just adding what made the accepted answer make sense to me, x^2 u^34 dx = C(u^34 du) where C is a constant because I wanted to emphasize the general form. I was helping a friend with homework ...
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Antiderivitive of $\sqrt[4]{a-x^4}$? [closed]

What is the antiderivitive of $\sqrt[4]{a-x^4}$ where $a$ is a constant?
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3 answers
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Can someone help find methods to study the asymptotic behavior of these integrals as $\rho \to \infty$?

I want to know methods to study the asymptotic behavior as $\rho \to \infty$ of the following integrals where $\rho>0$ and $q \geq 0$: $$ \int_{-1}^{1} e^{-\rho t} (1-t^2)^{q-1/2}dt$$ $$ \int_{1}^{\...
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1 answer
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Computing areas using Green's theorem

I want to compute the area of the surface $B$ with boundary parametrised by $$ \gamma(t)=\left(\begin{array}{c} \sin t \\ 4 \cos ^{2} t+\cos t \end{array}\right), \quad t \in[0,2 \pi] $$ ...
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Solving integral with generic pdf

I have a continuously differentiable function $h:[\underline{x},\overline{x}] \rightarrow \mathbb{R}$, and a continuous random variable $X$ distributed according to a cdf $F$, with full support on $[\...
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Difference of modified Bessel functions in integral form

I know the following integral representation of modified Bessel functions of first kind: $$I_q(\rho) = \frac{\left(\frac{\rho}{2}\right)^q}{\Gamma(q+1/2)\Gamma(1/2)} \int_{-1}^{1}e^{-\rho t}(1-t^2)^{q-...
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3 votes
1 answer
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My teacher said that this is not necessary in the line integral, but why?

Question: Calculate the Scalar line integral: $$\int_C \left(𝑥\,𝑑𝑥 − 𝑦\,𝑑𝑦\right)$$ C is the segment traveled in the direction: $$(1,1)\,to\,(2,3)$$ I started by solving this question by ...
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2 votes
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Understanding when we need injectivity requirement for substitution in integrals and when not and why

I am expanding my query i asked before so as to get a more well explained reasoning for this :. Suppose we are required to evaluate three integrals : $\int_{0}^{4} \frac{2x-4}{x^2 - 4x + 5}dx$ $\...
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Find the standard integral table of Beta function containing $n!$

I want to find that \begin{align*} \int_{0}^{\infty} \frac{(t)^n}{(1+t)^{2j+2}}dt = \sum_{n=0}^{2j} [\frac{(2j- n)! n!}{(2j+1)!}] \end{align*}\\ I have got ...
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Does decreasing the integration step always lead to a more accurate solution? Are there examples where this is not the case? [duplicate]

I understand that decreasing the integration step can increase the accuracy of the solution using certain methods. I imagine that this is not always the case though. Are there any methods for which ...
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0 votes
1 answer
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Changing limit for a dummy variable

I am given an integral: $$g(\text{z})=\int_0^z \xi ^3 e^{-3 A (\xi) -B^2 \xi ^2} \, d\xi$$ I wish to shift the coordinate from $z$ to ${r}$ with $z=\frac{1}{r}$. How to change the limits for the dummy ...
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1 vote
1 answer
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$\lim\limits_{n\to\infty}\int_0^2 \frac{nx^{n-1}e^{-x^n}}{1+x}dx=1-\lim\limits_{n\to\infty}\int_0^2\frac{e^{-x^n}}{(1+x)^2}dx$

How can I show $\lim\limits_{n\to\infty}\int_0^2 \frac{nx^{n-1}e^{-x^n}}{1+x}dx=1-\lim\limits_{n\to\infty}\int_0^2\frac{e^{-x^n}}{(1+x)^2}dx$? It is $\frac{d}{dx}x^n=nx^{n-1}$ so I tried to solve it ...
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Prove trat $\exists c>0$ that satisfies the inequality [duplicate]

Consider the real Hilbert space $H_0^1(0,1):=\{u:[0,1]\to\mathbb{R}$ absolutely continuous : $u'\in L^2(0,1), u(0)=u(1)=0\}, \langle u,v \rangle:=\int_0^1 u'v'$ $(i)$ Prove trat $\exists c>0$: $$||...
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Understanding why even though derivative exists but the derivative is not integrable in the given x range

Why is the derivative which exists everywhere for $f(x) = x^2 \sin(\frac{1}{x^2})$ for $x \neq 0$ and $f(x) = 0$ for $x=0$ is not integrable over a region $[a,b ] \in \mathbb{R}$ having $0$ included ...
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Let $X$ be a random variable following the standard cauchy distribution. Show that $E[X^{\alpha}]$ exists, $\forall\alpha\in(0,1)$

Let $X$ be a random variable following the standard cauchy distribution. Show that $E[X^{\alpha}]$ exists, $\forall\alpha\in(0,1)$ According to me, $E[X^{0.5}]=\int^{\infty}_{-\infty}\dfrac{\sqrt{x}}{...
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Complex Integral $\int^\pi_0 x\cdot \overline{\sin(nx)}dx$

I was solving the following problem: $$\langle f, e_n \rangle = \int^\pi_0 f\cdot \overline{e_n}\ dx=\int^\pi_0 x\cdot \overline{\sqrt{\frac{2}{\pi}}\sin(nx)}\ dx$$ $$\sqrt{\frac{2}{\pi}}\int^\pi_0 x\...
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2 answers
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Convergence of the following improper integral

I want to study the convergence of $\displaystyle\int_0^1 \frac{dx}{x^p-x^q}$ I noticed I could assume $p>q$, and I saw that in order for it to converge it must be $q<1$. I am not able to study ...
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Find the value of $\int_{1}^{4}\int_{\max\{\frac{1}{y},\frac{y}{2}\}}^{\min\{\frac{8}{y},{y}\}} \frac{y}{x}\sin(\frac{y}{x}+xy)dxdy$

$$\int_{1}^{4}\int_{\max\left\{\frac{1}{y},\frac{y}{2}\right\}}^{\min\left\{\frac{8}{y},{y}\right\}} \frac{y}{x}\sin\left(\frac{y}{x}+xy\right)dxdy$$ I'm trying to find this value through integration ...
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2 votes
1 answer
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Why do we use exponentials while integrating trigonometric functions in complex analysis

Let p(x) be some polynomial function. Now, we have an integral of the form : $$I=\int_{-\infty}^{\infty} \frac{\cos(x)}{p(x)}dx$$ What is usually done is that, we define this integral as : $$I'=\int_{-...
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2 votes
0 answers
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Expressing $\int_0^\pi \sin^n(x)dx$ in terms of the gamma function

Let $I_n = \int_0^\pi \sin^n(x)dx$ and suppose that we have already established a recursive relation $I_n = \frac{n - 1}{n}I_{n-2}$ and we know that $\Gamma(x + 1) = x\Gamma(x), \Gamma(1/2) = \sqrt{\...
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7 votes
4 answers
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How to evaluate $\int^{\infty}_0 \frac{x^{1010}}{(1 + x)^{2022}} dx$?

How to evaluate the following integral? $$\int^{\infty}_0 \frac{x^{1010}}{(1 + x)^{2022}} dx$$ Here's my work: $$I = \int_0^\infty \dfrac{x^{1010}}{(1+x)^{2022}} dx = \int_0^\infty \dfrac{1}{x^{1012}(...
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2 answers
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Evaluate $\int_{\mathcal{C}}\frac{e^z}{z^2}\,\mathrm{d}z$

How can I solve this problem using Cauchys integral formula $$\int_{\mathcal{C}}\frac{e^z}{z^2}\,\mathrm{d}z$$ When $$C:|z| = 1$$ I believe I should get something like this$$\int_{|z|=a} \frac{f(z)}{z-...
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2 votes
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Expressing the convolution integral of scaled and translated arguments

I was reviewing the Fourier transform where I came across the convolution integral. If the convolution $x(t)*y(t) = \int_{-\infty}^{\infty}x(\tau)y(t-\tau)d\tau$ be defined, how the convolution of $x(...
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1 answer
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can the upper integral of a function be larger than its domain? [closed]

When looking at Darboux integrals, is it possible for the upper integral to be larger than the product of the length closed interval and the range?
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Integrating $3 \cos^2 x \sin x$ [closed]

$$I = \int_0^1 3 \cos^2 x \sin x \,{\rm d}x$$ Please show it step by step and explain each step.
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4 votes
3 answers
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If $S$ is set of lower sums for partitions having $n$-equal subintervals, does $\sup S$ equal the $\sup$ of the set of lower sums for all partitions.

Firstly, here are two relevant definitions: Let $a \lt b$. A partition of the interval $[a,b]$ is a finite collection of points in $[a,b]$, one of which is $a$ and one of which is $b$. Suppose $f$ ...
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1 vote
0 answers
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Log-normal distribution: Why is this a density function?

I want to prove that $$f(x) = \frac{1}{\sigma x \sqrt{2\pi}} \exp(-\frac{\ln(x)^2}{2\sigma^2})$$ for $x>0$ and $f(x)=0$ for $x\leq 0$ is a density function, with $\sigma > 0$. So what I have to ...
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Trouble understanding a criteria for integrability

Let $f : [0, 1] → \mathbb{R}$ be bounded and $f ≥ 0$. Prove that if the set $\{x\in[0,1]|f(x)\geq \lambda \}$ is finite for every $λ > 0$ then $f$ is integrable. I'm having trouble understanding ...
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1 vote
1 answer
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Integrating distributions over submanifolds

Distributions may be "integrated against" (i.e. evaluated on) test functions, the following notation (for $T$ a distribution) is fairly common: $$ T(f) = \int T(x) f(x)\, \mathrm{d}x $$ I'm ...
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Relation between normalized product of integrals of two functions and integral of product of two normalized functions

Could someone let me know if there is any relation between A and B, if $A=\frac{\int^L_0 f(x)g(x)dx}{L}$ and $B=\frac{\int^L_0 f(x)dx\cdot\int^L_0 g(x)dx}{L^2}$
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Prove that the integral $\int _0^{\frac{\pi }{2}}\:\frac{1}{\sin\left(x\right)+\cos\left(x\right)}dx$ is bounded [closed]

I need to prove the following claim without solving the integral, can anyone help me with this? $$\frac{\pi }{2\sqrt{2}}\le \int _0^{\frac{\pi }{2}}\:\frac{1}{\sin\left(x\right)+\cos\left(x\right)}dx\...
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0 votes
2 answers
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How to interpret integrals that have conditions written beside them

sorry if this question has been asked before. I tried finding similar questions but couldn't find any. I have very little background in statistical mechanics, but I have been reading some literature, ...
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3 votes
2 answers
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Find $\displaystyle \int \dfrac{dx}{(x^2 + a^2)^3} $

I would like to find the anti-derivative $ \displaystyle \int \dfrac{dx}{(x^2 + a^2)^3 }$ My attempt: By substitution: $ x = a \tan(\theta) \Rightarrow dx = a \sec^2(\theta) d\theta$ Then the ...
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1 vote
2 answers
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How to Evaluate $\int_1^4 (\frac{1}{2t}+i)^2 dt$

PROBLEM$$\int \limits _1^4\left (\frac{1}{2t}+i \right )^2\,dt.$$How can I solve this? I believe its an indefinite integral and I can probably expand it by using $(a+b)^2=a^2+2ab+b^2$ to get something ...
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Is there a probabilistic concept or theory for infinitesimal logarithmic product interpretation of integral?

If we have a number of independent events in probability, we can calculate it's likelihood : $$\prod_{\forall i} p_{i}$$ We can also consider ( where $H$ is the Heaviside step function ) $$\int L(t) H(...
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2 votes
0 answers
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How was the precession of Mercury's orbit calculated in the days before computers?

Over the weekend, I wrote a little program that simulates the solar system (or any set of bodies). I use off-the-shelf orbital elements to set the starting conditions and then do a simulation of ...
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3 votes
1 answer
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Confused what I did wrong for $\int_{0}^{\infty} \frac{1}{1 + x^4} \, dx$

I did $ x = u\sqrt{i}$ $$\sqrt{i}\int_{0}^{\infty} \frac{1}{1 - u^4} \, du$$ $$\sqrt{i}\int_{0}^{\infty} \frac{1}{1 - u^2} \cdot \frac{1}{1 + u^2} \, du$$ $ v = \tan^{-1}(u)$,$dv = \frac{1}{1 + u^2} ...
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1 vote
2 answers
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$\iint_Sz^2d\sigma$ where $S$ is an area of the cone $z=\sqrt{x^2+y^2}$ between planes $z=0$ and $z=1.$

What is the value of $\iint_Sz^2d\sigma$ where $S$ is an area of the cone $z=\sqrt{x^2+y^2}$ between planes $z=0$ and $z=1.$ To solve this by using polar integration I think the integral transforms ...
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1 answer
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Indication: $\displaystyle \lim_{x \to \infty}{\int^{x}_{1}{\ln(f(t))dt}}$

Let $f$ be a function $f:\mathbb{R}\to\mathbb{R}$, $f(x)=4x^3 + 1$. I have to find out $\displaystyle \lim_{x \to \infty}{\int^{x}_{1}{\ln(f(t))dt}}$. My intuition is telling me the limit is $+\infty$....
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1 vote
1 answer
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Using Feynman's technique to find $\int_0^{\infty}\frac{\cos{x}}{x^2+1} dx$

Using Feynman's technique, I want to evaluate $$\int_0^{\infty}\frac{\cos{x}}{x^2+1} dx$$ I set $$I(a)=\int_0^{\infty}\frac{\cos{ax}}{x^2+1} dx$$ which gives $$I'(a)=\int_0^{\infty}\frac{-x\sin{ax}}{...
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0 votes
0 answers
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Integrating the increments of a Poisson Process [closed]

In examples 11.4.4, how exactly are we able to say that the answer to the integration will be 0? Φ(s) takes the value of either 0 or 1.
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-3 votes
0 answers
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Compute an integral with modulus [closed]

Let $0\leq r < 1$, How do I suppose to do the next integral? $$ \int_{-\pi}^{\pi}\dfrac{1}{|1-re^{i\theta}|^2}d\theta. $$
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-3 votes
0 answers
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Let $R = [a, x] \times [b, y]$, and let $f : R\to\Bbb R$ be a function of class $C^1$ Prove that $F$ is of class $C^2$ in $R$ [closed]

Let $R = [a, x]\times [b, y]$, and let $f :R\to\Bbb R$ be a function of class $C^{1}$. If $F : R → R$ is given by $F(x,y)=\int \int_{R} f(x,y)dA$ Prove that $F$ is of class $C^{2}$ in $\operatorname{...
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0 votes
0 answers
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Checking a 1-Lipschitz condition for Normal density integral [closed]

Let $\phi$ denote the density function for a $N(0,1)$ random variable. Is the following true? $$\int_{\mathbb{R}} \vert \phi(x)-\phi(x-\theta)\vert dx = O(\theta) $$ as $\boldsymbol{\theta \rightarrow ...
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