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Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

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Positiveness of sinc function integral

Is there a simple way to see that $$ \int_{-a}^0 \int_0^a \frac{\sin(x-y)}{x-y} \, dx \, dy \geq 0 $$ for all $a>0$?
lasik43's user avatar
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2 votes
1 answer
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$\int_0^{\pi} ((\sin{x})^3+(\cos{x})^3)^n dx$ is rational iff $n$ is odd

Define the integral $$I_n = \int_0^{\pi} ((\sin{x})^3+(\cos{x})^3)^n dx$$ for any natural number $n$. I am trying to show that $$I_n\text{ is rational } \iff n \text{ is odd}$$ My first idea was to ...
Sam's user avatar
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1 answer
63 views

Integral of $e^{\frac{-1}{x(1-x)}}$

The function $e^{\frac{-1}{x(1-x)}}$ is a smooth positive function supported on the compact set $[0,1]$. What is its integral? (According to WolframAlpha, it's approximately 0.007, but I was hoping ...
Ryan's user avatar
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4 votes
0 answers
128 views

I'm stuck on this integral $\int_{0}^{1} \frac{\ln (x) \ln^{2}(1+x) \ln(1-x)}{x} \ dx$

I was trying to evaluate this famous integral $$\int_{0}^{1} \frac{\ln (x) \ln^{2}(1+x) \ln(1-x)}{x} \ dx $$ Here is my attempt so solve the integral \begin{align} &\int_{0}^{1} \frac{\ln (x) \ln^{...
aryan's user avatar
  • 121
3 votes
0 answers
77 views

how to evaluate $\int_0^{\infty} \frac{x \ln ^2\left(1-e^{-2 \pi x}\right)}{e^{\frac{\pi x}{2}}+1} d x$

Question: how to evaluate $$\int_0^{\infty} \frac{x \ln ^2\left(1-e^{-2 \pi x}\right)}{e^{\frac{\pi x}{2}}+1} d x$$ MY try to evaluate the integral $$ \begin{aligned} & I=\int_0^{\infty} \frac{x \...
aryan's user avatar
  • 121
1 vote
1 answer
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Application Fundamental Theorem of calculus - Replacing a function with its integrals

I was going through the solution of one of the questions asked on this platform. Question link I was able to follow the accepted answer until the last step where @Aryabhata substitutes $$ g(t) = g(0) +...
Ark's user avatar
  • 81
2 votes
1 answer
66 views

Determine whether $ \int_{2}^{\infty}\frac{\cos\left(\ln x\right)}{\left(x+1\right)\ln x}dx $ diverges or converges.

I'm going through my learning material, and one of following question at the end of it was: Determine whether the integral $ \int_{2}^{\infty}\frac{\cos\left(\ln x\right)}{\left(x+1\right)\ln x}dx $ ...
Emanuel L's user avatar
4 votes
1 answer
109 views

Separation problem in definite integral of piecewise function

My question is about separation of $\displaystyle \int _0 ^2 f(x) dx$ defined by $f(x) = \left\{\begin{aligned} &x^2 ,\ x \in [0,1]\\ &x^4+4 ,\ x \in (1,2] \end{aligned} \right.$ Of course, we ...
Fuat Ray's user avatar
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4 votes
1 answer
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closed form of $\int_{[0 ; 1]} \frac{\operatorname{Li}_3\left(-x^2\right)}{1+x} d x$

Question: closed form of $$\int_{[0 ; 1]} \frac{\operatorname{Li}_3\left(-x^2\right)}{1+x} d x$$ My try to solve the integral $$ \begin{aligned} & I=\int_{[0 ; 1]} \frac{\operatorname{Li}_3\left(-...
aryan's user avatar
  • 121
1 vote
2 answers
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$\int_0^\pi \frac1{1-x\cos \theta-y\sin \theta}+\frac1{1+x\cos \theta+y\sin\theta}d\theta=\frac{2\pi}{\sqrt{1-x^2-y^2}}$

My question is: how can we prove the following integral equation? For any $x,y\in \mathbb R$ such that $x^2+y^2<1$, $$ \int_0^\pi d\theta\left(\frac1{1-x\cos \theta-y\sin \theta}+\frac1{1+x\cos \...
Mike Earnest's user avatar
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1 vote
1 answer
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Diverging integrals and bounds

My analysis is quite rusty, so I am looking for some help with the following: Suppose we have a function which diverges (logarithmically) at $x = 0$, e.g. $f(x) =\frac{1}{x}$. If we then want to ...
Geigercounter's user avatar
3 votes
1 answer
70 views

Help with Evaluating a Definite Integral Involving Nested Radicals

I'm working on a calculus problem and need help solving the following definite integral: I'm struggling to simplify the integrand or find a substitution that makes the integral easier to evaluate. ...
Xbz-24's user avatar
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8 votes
1 answer
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Finding $\left[\left(\frac{A^3-B^3}{B^3}\right)+\left(\frac{A^4+B^4}{B^4}\right)\right]$

Question: If $A=\displaystyle\int_0^1\left(1-x^{2023}\right)^{\frac 1{2022}}\, dx$ $\quad$and $\quad$$B=\displaystyle\int_0^1\left(1-x^{2022}\right)^{\frac 1{2023}}\, dx \quad$ then find the value of $...
Skdmg's user avatar
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4 votes
3 answers
159 views

Closed form for $\int_0^{\pi/2}\arctan\left(\frac12\sin x\right)\mathrm dx$?

Is there a closed form for $I=\int_0^{\pi/2}\arctan\left(\frac12\sin x\right)\mathrm dx$ ? Context Earlier I asked "Find the area of the region enclosed by $\frac{\sin x}{\sin y}=\frac{\sin x+\...
Dan's user avatar
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How do I integrate this? Problem with derivative and integration [closed]

I want to find the length of an arc given the equation by integration. formula: https://i.stack.imgur.com/45Zty.png The equation of arc: $y = 2 * (1 - x^{(\frac{2}{3})})^{(\frac{3}{2})};$ graph limits:...
Loly18's user avatar
  • 15
1 vote
1 answer
89 views

How do I integrate this? Problem (error) with integration [closed]

I need to find the length of an arc given the equation by integration. formula equation of arc: $$f(x) = \sqrt{\frac{1}{16}-\left(x-\frac{1}{4}\right)^2}$$ I got the derivative of $f'(x) = \frac{-4x+1}...
Loly18's user avatar
  • 15
2 votes
3 answers
142 views

Evaluating $\lim_{t\to\infty}\left(\left(\log\left(t^2+\frac1{t^2}\right)\right)^{-1}\int_1^{\pi t}\frac{\sin^25x}{x}dx\right)$

The value of $$\lim_{t\to\infty}\left\{\left[\log\left(t^2+\frac1{t^2}\right)\right]^{-1}\int_1^{\pi t}\frac{\sin^{2}\left(5x\right)}{x}\,{\rm d}x\right\} $$ is ______ (rounded off to two decimal ...
Ketan Choudhary's user avatar
2 votes
0 answers
35 views

Solving $\int_0^1 \frac{\log^2(1\pm x)}{x}\mathrm{d}x$ [duplicate]

I want to solve the integral $$\mathcal{I}=\int_0^1 \frac{\log^2(1-x)}{x}\mathrm{d}x$$ I tried solving it by twice differentiating the Beta function to obtain the logarithm found in $\mathcal I$ and ...
mira666's user avatar
  • 43
3 votes
1 answer
74 views

How to approximate $\frac{(2n+1)}{2^n} \int_0^1 \left( 1-x^2+\sqrt{1+2x^2-3x^4}\right)^ndx$

Let $$ f(x) = \frac{1-x^2+\sqrt{1+2x^2-3x^4}}{2} $$ How to approximate the integral $$ I_n = (2n+1)\int_0^1 f(x)^n dx? $$ Experiments seem to indicate that it is something like $cn^{0.75}+1$ where $c$ ...
ploosu2's user avatar
  • 8,886
1 vote
1 answer
36 views

Show that $\frac{2}{9} \leq \frac{\tan(\frac{2}{\sqrt{5}})}{\sqrt{5}} \leq \frac{2}{5}$

I have to show that $\frac{2}{9} \leq \int^1_{-1} \frac{1}{x^2+2x+6}dx \leq \frac{2}{5}$ . $\int^1_{-1} \frac{1}{x^2+2x+6}dx= \int^1_{-1} \frac{1}{(x+1)^2+5}dx$ Substituting $u=x+1 , du=dx ,$ upper ...
Antony Theo's user avatar
3 votes
2 answers
254 views

Error with Cauchy Integral Formula

I have been told to calculate $$ \int_0^{2\pi}\frac{1}{2+2\text{sin}(\theta)} d\theta $$ I set $z = e^{i \theta}$ so parameterising by the unit circle and ended up with $$ \int_C \frac{1}{(z+i)^2} dz $...
adisnjo's user avatar
  • 205
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0 answers
17 views

Volume under a cone and above a disk

To find the volume under the cone $$z=\sqrt{x^2+y^2}$$ and above the disk $$x^2+y^2\leq16$$ I used cylindrical coordinates as follows: $$x=r\cos(\theta),\qquad y=r\sin(\theta),\qquad z=z$$ with $0\leq ...
mvfs314's user avatar
  • 2,029
0 votes
1 answer
19 views

Non-Uniform Partition Integrals

What is the specific branch of mathematics that focuses on integrals with non-uniform partitions? I'm working on solving integrals of this type: $$\int_{a}^{b}f(dx)$$ Alternatively, how is the ...
Taller de Hidrodinmica Faculta's user avatar
2 votes
1 answer
77 views

Evaluating the sum $\sum_{n=1}^{\infty}{\frac{(-1)^n n}{n^2+1}}$

I am trying to solve the integral \begin{align} \int_0^\infty \frac{\cos x}{e^x +1}\mathrm{d} x \end{align} and I have encountered this sum: \begin{align} \sum_{n=1}^{\infty}{\frac{(-1)^n n}{n^2+1}} \...
mira666's user avatar
  • 43
0 votes
0 answers
31 views

The integral of $\int_0^1 x^a(1+x)^b(1-x)^c e^{-dx^2} dx$

As the title mentioned, I want to get a closed-form result of $\int_0^1 x^a(1+x)^b(1-x)^c e^{-dx^2} dx,$ where $d>0$, $a,b,c$ may be very large numbers. A related integral is $\int_0^1 x^a (1-x)^b ...
jobs adam's user avatar
1 vote
1 answer
61 views

How to evaluate $\int_1^{\infty}\frac{t^2\ln^2 t\ln(t^2-1)}{1+t^6}{\rm d}t $

I was evaluating Evaluate $\displaystyle\int_0^{\infty} x^2\ln(\sinh x)\operatorname{sech}(3 x){\rm d}x .$ On the path of integrating the main function, I am stuck at this integral. I don't know how ...
Sbsty 's user avatar
0 votes
1 answer
35 views

Volume of a Cone using Change of Variables

I am having trouble with the following problem. Let $a,b,c>0$ and let $\Omega$ be the region enclosed by the cone, $\frac{x^2}{a^2}+\frac{y^2}{b^2} =\frac{z^2}{c^2}, 0\leq z\leq c $. I'm tasked to ...
100fcb's user avatar
  • 65
2 votes
3 answers
123 views

How to integrate $\int_{0}^{1}\frac{x^{100}}{1+x}dx$?

Q) How to integrate $$\int_{0}^{1}\frac{x^{100}}{1+x}dx$$ ? The only process which I know is by substituting $(1+x)$ as $u$ and then $dx$ becomes $du$. Therefore,finally the integral will be $$\int_{0}...
Dropper's user avatar
  • 67
18 votes
2 answers
272 views

What is minimum of the integral function $I(x)= \int_0^\infty \frac{1}{(1+t^x)^x} \,dt$

A while ago I stumbled on a YT movie with regards the integral $$\int_0^\infty \frac1{(1+x^\phi)^\phi} dx = 1 .$$ Here $\phi=1,6180\ldots$ is the Golden Ratio. But then I thought, what will happen is ...
Arthur van Houdt's user avatar
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2 answers
53 views

Finding the area of the curved figure bounded by $y=x^{2/3}-1$, $y = 8$, and $y = 0$

How do you find the area of the curved figure defined by the graphs of these relations? $$y = x^{2/3} - 1$$ $$y = 8$$ $$y = 0$$ Regarding this assignment. I realised where the figure is defined: from ...
Loly18's user avatar
  • 15
2 votes
1 answer
64 views

Integral of a rectified power law with Gaussian noise for non-integer powers

I am interested in the following integral: $$f(x)=\int dt \phi(t) [x+\sigma t]_+^n $$ where $\phi(x)=e^{-x^2/2}/\sqrt{2\pi}$ is a standard Gaussian distribution, $[x]_+=max(x,0)$ is a rectification, $...
Uri Cohen's user avatar
  • 375
3 votes
3 answers
172 views

Show that $\int_{0}^{1} \frac{\tan^{-1}(x^2)}{\sqrt{1 - x^2}} \, dx = \frac{1}{2}\pi \tan^{-1}\left(\sqrt{\frac{1}{\sqrt{2}} - \frac{1}{2}}\right)$

Problem: Show that $$\int_{0}^{1} \frac{\tan^{-1}(x^2)}{\sqrt{1 - x^2}} \, dx = \frac{1}{2}\pi \tan^{-1}\left(\sqrt{\frac{1}{\sqrt{2}} - \frac{1}{2}}\right)$$ Some thinking before trying At least we ...
Sbsty 's user avatar
7 votes
1 answer
177 views

Compute the integral: $\int_{1}^{\infty}\frac{2+\sqrt{x+1}}{(1+x)^2-\sqrt{x-1}}dx$

The integral is from my MIT Integration Bee 2025 Mock Training Problems. But unfortunately, I forgot the solutions nor the trick to this problem. I tried letting $u=\sqrt{x-1}$, then perform trig-...
Silver's user avatar
  • 747
-1 votes
1 answer
46 views

Can anyone explain how this question works, also what limit gives exact value of 1+ln 2 [closed]

enter image description here I have tried to use the sum of n terms limit thing....but I can't understand how the limit works I used LHS = $\int_0^1 (x+2)/(x+1)$ but this is the same case for both ...
Zayden's user avatar
  • 1
1 vote
0 answers
93 views

integral of $\int_0^\infty \frac{2y^n}{e^{2y} - 1} dy$

I'm reading the David Tong's notes (p.111) from Cambridge and I'm facing an integral which I don't know how the author solve it. $$\int_0^\infty \frac{2y^n}{e^{2y} - 1} dy = \frac{I_n}{2^n}$$ $$I_n = ...
epselonzero's user avatar
5 votes
0 answers
131 views

Finding a closed form for $ \int_0^1 \frac1x \ln\left(\frac{\ln\left(\frac{1-x}{2}\right)}{\ln\left(\frac{x+1}{2}\right)}\right)\, \mathrm{d}x $

I want a closed form for the following integral $$ \int_0^1 \frac1x\;\ln\left(\frac {\ln\left(\frac{1-x}{2}\right)}{\ln\left(\frac{x+1}{2}\right)}\right)\, \mathrm{d}x $$ An integration by parts ...
Jessie Christian's user avatar
1 vote
0 answers
48 views

Show equivalence of two very long integrals.

I'm trying to show that the following integral $$\int_0^\infty \int_{-\infty}^s f(x \vee s-b) \sqrt{\frac{2}{\pi}}\frac{1}{t^{\frac{3}{2}}}(2s-b)e^{-\frac{(2s-b)^2}{2t}-\mu(b+\frac{\mu t}{2})}db ds$$ ...
user82832's user avatar
-2 votes
0 answers
35 views

Product of two improper integals who converge is divergent [closed]

Give an example of two functions $f(x)$ and $g(x)$ such that their integrals both converge respectively, on $[1, \infty]$, but their product integral diverges.
Nethanel 's user avatar
0 votes
0 answers
33 views

Evaluating $4 \sqrt{17}\int_0^1 t e^{-t}dt$ [closed]

I'm not sure how to evaluate this integral. $$4 \sqrt{17}\int_0^1 t e^{-t}dt$$ I know it's supposed to be done by parts, with $u = t$ and $dv = e^{-t}$, but I keep getting $4 \sqrt{17}( -2e^{-1} +1)$ ...
Alec Tang's user avatar
0 votes
5 answers
82 views

Help with integration over a triangular region.

I'm currently trying to wrap my head around double integrals over a triangular region, when you are given the vertices of the triangle. I need to do the integral $$ \iint_D 2e^{-y-x} \,\mathrm{d}y\,\...
HMND's user avatar
  • 1
3 votes
4 answers
140 views

Where is the error in evaluating this integration? :$\int_0^{\frac{\pi}{2}} \frac{dx}{\cos(x)+\sin(x)}$ [duplicate]

Where is the error in evaluating this integration? :$$\int_0^{\frac{\pi}{2}} \frac{dx}{\cos(x)+\sin(x)}$$ let: $ u=\sin(x)+\cos(x)$ this implies $u^2=1+\sin(2x)\implies dx=\frac{u}{\sqrt{2u-u^2}}du$ ...
Mostafa's user avatar
  • 1,570
0 votes
1 answer
48 views

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
5 votes
2 answers
111 views

How to evaluate$\lim\limits_{n\to\infty}n\left(n\left(n..\left(\int_0^1\left(\frac{\sqrt[n]{x}+1}{2}\right)^ndx-l_0\right)-l_1..\right)-l_{m}\right)$?

$$\lim_{n \to \infty} \int_0^1 \left( \frac{\sqrt[n]{x}+1}{2} \right)^n dx= \frac{2}{3}$$ I became curious what happens If we do the following: $$\lim_{n \to \infty} n\left(\int_0^1 \left( \frac{\...
pie's user avatar
  • 4,514
0 votes
1 answer
58 views

How to prove that $\left( \frac{\sqrt[n]{x}+1}{2} \right)^n $ converge uniformly on $[0, 1-\delta]$?

I saw this problem: $$\lim_{n \to \infty} \int_0^1 \left( \frac{\sqrt[n]{x}+1}{2} \right)^n dx $$ Assuming the limit and integral could be interchanged then we have the limit is $\frac{2}{3}$. I don't ...
pie's user avatar
  • 4,514
7 votes
0 answers
212 views

How did Ramanujan find $\sum_{n=0}^\infty (-1)^n\frac{(1/2)_n(1/4)_n(3/4)_n}{n!^3}\frac{644n+41}{25920^n}=\frac{288\sqrt{5}}{5\pi}?$

The formula $$\sum_{n=0}^\infty (-1)^n\frac{(1/2)_n(1/4)_n(3/4)_n}{n!^3}\frac{644n+41}{25920^n}=\frac{288\sqrt{5}}{5\pi}$$ (in older notation) appears as eq. 38 in Ramanujan's paper Modular equations ...
Nomas2's user avatar
  • 635
0 votes
0 answers
58 views

How to compute $\displaystyle\int_{i\infty+1}^1\frac{e^{i2(n+1)x}}{\sin(x)}\, \mathrm dx$

I need to solve $$\int_{i\infty+1}^1\frac{e^{i2(n+1)x}}{\sin(x)}\, \mathrm dx$$ which appears in the closed form of $$\sum_{k=0}^n\frac{e^{i2k}}{2k+1}.$$ By the looks of the integrand, Euler's ...
Conreu's user avatar
  • 1,918
1 vote
1 answer
47 views

Integrate a sum of trig function under absolute value

Let $n \in \mathbb{N}$, I'm trying to compute en explicit formula for the following integral: $$ \operatorname{I}\left(n\right) = \int_{\left[0,2\pi\right]^{n}} \left\vert\cos\left(x_1\right) + \cdots ...
MathRevenge's user avatar
2 votes
2 answers
197 views

Help needed with an integral $\int_0^1 \frac{\ln(x) \ln(1+x) }{1+x^2} dx$.

I am trying to evaluate the integral $$ \int_0^1 \frac{\ln(x) \ln(1+x) }{1+x^2} dx $$ Integration by parts gives \begin{eqnarray*} \int_0^1 \frac{\ln(x) \tan^{-1}(x) }{1+x} dx + \int_0^1 \frac{\ln(x) ...
Donald Splutterwit's user avatar
6 votes
1 answer
231 views

Evaluate $\int_0^\infty\frac{dx}{1+x^2}\prod_i\arctan a_ix$ (product of arctangents and Lorentzian)

Define $$I(a_1,\dots,a_n)=\int_0^\infty\frac{dx}{1+x^2}\prod_{i=1}^n\arctan a_ix$$ with $a_i>0$. By this answer $\newcommand{Li}{\operatorname{Li}_2}$ $$I(a,b)= \frac\pi4\left(\frac{\pi^2}6 -\Li\...
Parcly Taxel's user avatar
3 votes
2 answers
159 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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