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Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

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1answer
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Compute $\lim\limits_{x \to +\infty}\dfrac{\ln x}{ \int_0^x \frac{|\sin t|}{t}{\rm d}t}$.

Problem Compute $$\lim\limits_{x \to +\infty}\dfrac{\ln x}{\displaystyle \int_0^x \dfrac{|\sin t|}{t}{\rm d}t}.$$ Comment Maybe, we can solve it by L'Hospital's rule, but there still exists a ...
0
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2answers
29 views

Prove $\int^{\infty}_{0} \frac{\cos(ax)}{\cosh(\beta x)}dx = \frac{\pi}{2\beta}\operatorname{sech}(\frac{a\pi}{2\beta})$

Proof of 3.981.3 Gradshteyn ed.8. $$\int^{\infty}_{0} \frac{\cos(ax)}{\cosh(\beta x)}dx = \frac{\pi}{2\beta}\operatorname{sech}(\frac{a\pi}{2\beta})$$ I was interested in the derivation (not ...
4
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0answers
38 views

$I = \int_0^k z^{m_1 - 1} \ln(1 + z) \left(\frac{m_1 z}{a} + \frac{m_2}{b} \right)^{-(m_1 + m_2)} \mathrm dz.$

Question: How to find the closed-form solution for the given integral? $$I = \int_0^k z^{m_1 - 1} \ln(1 + z) \left(\dfrac{m_1 z}{a} + \dfrac{m_2}{b} \right)^{-(m_1 + m_2)} \mathrm dz,$$ where $k, a, b,...
2
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1answer
74 views

Evaluate $\int^{2}_{0}\frac{\tan^{-1}(x)}{1+4x}\mathrm dx$

Evaluate $\displaystyle \int^{2}_{0}\frac{\tan^{-1}(x)}{1+4x}\mathrm dx$ My effort: \begin{align*} I(a)&=\int^{2}_{0}\frac{\tan^{-1}(ax)}{1+4x}\mathrm dx\\ I'(a) &= \int^{2}_{0}\frac{x}{(1+...
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2answers
50 views

Integration of exponential function $\int_{-\infty}^{+\infty} e^{\pi x^{2}}dx $?

Could you, please, help me calculate the integral $\int_{-\infty}^{+\infty} e^{\pi x^{2}}dx $?
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0answers
55 views

How to evaluate $\int_0^{\pi/2} \ln{\left(\arctan x\right)}\,dx$? [on hold]

How to solve $\displaystyle\int_0^{\frac{π}{2}} \ln{\left(\arctan x\right)}\,dx$?
2
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1answer
34 views

Is it ever feasible to actually pump water out of the top of a tank?

In math/calculus classes, a problem frequently posed asks how much work (J) is required to pump water out of the top of differently shaped water tanks. If you are unsure of what I am referring to, ...
2
votes
2answers
53 views

Complete Elliptic Integral of the First Kind Identity

Is there an identity for $\frac{K'(k)}{K(k)}=?$ where $K(k)=\int_0^{\frac{\pi}{2}}\frac{1}{\sqrt{1-k^2\sin^2(x)}}dx=\int_0^1\frac{1}{\sqrt{(1-t^2)(1-k^2t^2)}}dt$ is the Complete Elliptic Integral of ...
0
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1answer
65 views

Closed form for $\int_0^1 e^{\frac{1}{\ln(x)}}dx$?

I want to evaluate and find a closed form for this definite integral:$$\int_0^1 e^{\frac{1}{\ln(x)}}dx.$$ I don't know where to start. I've tried taking the natural logarithm of the integral, ...
0
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1answer
67 views

How to show that $\int_0^1x^{-x}dx = \sum_{n=1}^\infty n^{-n}$? [duplicate]

How would I go about showing that $\int_0^1x^{-x}dx = \sum_{n=1}^\infty n^{-n}$ Right now my numerical analysis class is covering gaussian quadrature but we have also covered interpolation. I'm not ...
2
votes
1answer
69 views

Calculate these integrals

I want to know how to calculate any of these integrals, which arise from computing the perimeter of the unit ball in the $p$-norm. $$\int_0^1(1+p^pt^{p(p-1)})^{1/p} dt;$$ $$\int_0^{\pi/2}(\cos(t)^{2-...
1
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1answer
40 views

Definite integration with e

So I was practising definite integration when I came across this one. $$\int_0^1 (\mathrm{e}x-x\mathrm{e}^x)\,\mathrm{d}x$$ So I calculated this to be: $$\int_0^1 \mathrm{e}x\,\mathrm{d}x - \...
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2answers
40 views

What's The Cylindrical coordinates for this $\int_0^6\int_{-\sqrt{6x-x^2}}^{\sqrt{6x-x^2}}\int_0^{6x-x^2-y^2}\left(x^2+y^2\right)dzdydx$

I want to convert this to cylindrical coordinates $$V=\int_0^6\int_{-\sqrt{6x-x^2}}^{\sqrt{6x-x^2}}\int_0^{6x-x^2-y^2}\left(x^2+y^2\right)dzdydx = 486π$$ I want to write it like this: $$V=\int_{\ }^{ ...
0
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2answers
41 views

Why is area under some symmetric curves zero and others not?

When finding the area under a curve, there appears to be a contradiction Like area under integral sinx from 0 to 2pi is zero because, the areas above and below cancel each other. But when finding ...
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0answers
72 views

Need help to compute the integral $\int_0^1\frac{x\cos x}{\sqrt{1+x}}\,\mathrm dx$ [on hold]

Need help to compute the integral$$\int_0^1\frac{x\cos x}{\sqrt{1+x}}\,\mathrm{d}x.$$
1
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1answer
55 views

Doubt on Integration

$$\int_0^4 \lfloor x/2 \rfloor \ d(x-\lfloor x \rfloor)$$ I don't get how we convert the given differential element into normal dx differential element. I plotted the graphs of $$\lfloor x/2 \...
6
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2answers
136 views

Solving $\sin(x) = \ln(x)$

While tutoring a student in introductory calculus, I inquired as to how she would set up an integral to determine the volume of the solid created by revolving the region bounded by the functions $\sin(...
3
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2answers
65 views

Evaluate $\int_0^1 \frac{1}{\sqrt x+\sqrt {(1-x)}}dx$

Evaluate $I=\int_0^1 \frac{1}{\sqrt x+\sqrt {(1-x)}}dx$. I applied $x=\sin^2\theta$,that makes $I=\int_0^{\pi/2} \frac{\sin2\theta}{\sin\theta+\cos\theta}d\theta$,but the further proceedings makes $I$...
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0answers
35 views

Airy integral with complex coefficients

In the book by Olivier Vallee and Manuel Soares about Airy function following integral has been given: $$ \int_{-\infty}^{\infty}e^{i(t^3/3+at^2+bt)}dt=2\pi e^{ia(2a^2/3-b)}\mathrm{Ai}(b-a^2) $$ I ...
1
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1answer
18 views

Finding the surface area of two solids of revolution

I have a couple of questions that are similar in nature: 1) I am trying to find the surface area of this when I rotate it around the x-axis. I have $y = \sqrt{5-x}$ when $3 \leq x \leq 5$ Say $dy/dx ...
0
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0answers
18 views

Cauchy integral with positive lower limit of integration

I’m trying to solve an integral of the following form: $\int ^b _a \frac{1}{(\omega^2+r_1)(\omega^2+r_2)}d\omega$ Where $r_1$ and $r_2$ are complex conjugates. I used the Cauchy theorem to solve ...
3
votes
2answers
72 views

Evaluate the integral: $\int_0^{\infty}\frac{\tan^{-1}(tx)}{x\left(1+x^2\right)} \mathrm{d}x$

I've been trying to evaluate the integral for a while now, and I've been unable to find it anywhere... I tried substituting $\tan^{-1}(tx)$ as $u$ but got nowhere... I have done dozens of other ...
0
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0answers
28 views

Rewriting a $\max\left\{0,\dots\right\}$ function in order to integrate the function more properly

Yesterday I asked a question about a certain integral. In the integral is the term: $$\max\left\{0,\left|\text{n}\cdot\sin\left(2\pi\cdot x\cdot t-\frac{\pi}{2}\right)\right|-2\cdot\text{z}\right\}\...
0
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2answers
57 views

Find the integral $\int_0^{\pi/2}\sin2x\sqrt{4\sin^4x+1}\,dx$ [on hold]

$\int_0^{\pi/2}\sin2x\sqrt{4\sin^4x+1}\,dx$. Give me a hint how to start, please. I have tried to integrate by parts, change variable, but it made it more complicated
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0answers
17 views

Definite integral of a given function.

How can I compute definite integral of the following function? $\int_{x(0)}^{0} \frac{dx}{k_2\,\sin x + k_1\frac{\cos x - 1}{\sin 2x}}$ $k_1$ and $k_2$ are positive constants. At this point, I know ...
3
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0answers
96 views

Closed form solution for $\int_0^1 \frac{\ln^2(x)\ln^2(1-x)}{x(1-x)} dx$ using Bose Integral

I'm looking for the solution to the following integral, but by using the Bose integral: $$\int_0^1 \frac{\ln^2(x)\ln^2(1-x)}{x(1-x)} dx$$ I got to this form when looking for a solution to the ...
5
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3answers
84 views

integral $C(a,b)=\int_0^{2\pi}\frac{xdx}{a+b\cos^2 x}$

I am looking for other methods to find the general integral $$C(a,b)=\int_0^{2\pi}\frac{xdx}{a+b\cos^2x}$$ To do so, I first preformed $u=x-\pi$: $$C(a,b)=\int_{-\pi}^{\pi}\frac{xdx}{a+b\cos^2x}+\pi\...
1
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1answer
67 views

A tricky limit involving exponential integrals

We define exponential integral according to https://en.wikipedia.org/wiki/Exponential_integral#Definition_by_Ein as $$\text{Ei}_n(x) = \int_{1}^{\infty} \frac{e^{-xt}}{t^n} dt$$ I'm trying to ...
0
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1answer
52 views

Prove that $0.4 ≤ \int_0^1 f(x) dx ≤ 0.5$ for $f(x) = x^{\cos x + \sin x} $. [duplicate]

Consider the function $$f(x) = x^{\cos x + \sin x} $$ defined for $x \ge 0.$ Prove that $$0.4 ≤ \int_0^1 f(x) dx ≤ 0.5.$$ I tried using max-min inequality but it didn't work.
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0answers
27 views

Finding an inverse Laplace transform of an integral that involves the $\max\left\{0,\dots\right\}$ function

I'm trying to tackle a complicated real world (electronics) question. In order to get the last part of the proof I need to find the following 'difficult' integral: $$\mathcal{L}_\text{s}^{-1}\left[\...
1
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1answer
41 views

Area between curves.

I have to calculate area bounded by curves : $(x^3+y^3)^2=x^2+y^2 $ for $ x,y \ge 0 $. I tried to use polar coordinates, but I have : $r^4(\cos^6\alpha +2\sin^3\alpha\cos^3\alpha + \sin^6\alpha)=1$
0
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1answer
33 views

Calculus Differential Equation

Problem: Let $f$ be a continuous function whose domain includes $[0,1]$, such that $0 \le f(x) \le 1$ for all $x \in [0,1]$, and such that $f(f(x)) = 1$ for all $x \in [0,1]$. Prove that $\int_0^1 f(...
4
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0answers
74 views

Evaluating a Polynomic-Trigonometric-Hyperbolic Integral

Within this AoPS thread it is asked to evaluate the following integral $$\mathfrak I~=~\int_0^\infty \frac{x\sin x}{\cos x+\cosh^2 x}\mathrm dx\tag1$$ In order to be precise there is also a ...
1
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1answer
51 views

How to see the convergence/divergence of this integral [duplicate]

I have a question, I want to know for which values of $p\in \mathbb{R}$ this integral have finite value: $$\int_{0}^{\infty} \frac{\sin{x}}{x^p}dx $$ I have shown that for $p=1$ the integral is finite ...
1
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2answers
100 views

Finding $\lim_{n \to \infty} \int_0^n \frac{dx}{1+n^2\cos^2x}$

Find $$\lim_{n \to \infty} \int_0^n \frac{dx}{1+n^2\cos^2x}$$ I tried: mean value theorem. variable change with $ \tan x = t $ but I need to avoid the points which are not in the domain of $\tan$ ...
0
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1answer
38 views

Finding $\lim_{n \to \infty} \int_{0}^{\pi/3} \frac{1}{1+\tan^n(x)}\,dx$

Compute $$\lim_{n \to \infty} \int_{0}^{\pi/3} \frac{1}{1+\tan^n(x)}\,dx$$ I tried to do a variable change $\tan x=t$ and arrived at another integral but I haven't solved it yet.
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1answer
29 views

Evaluating $\int_0^l (lx-x^2)\sin\left(\frac{n\pi x}{l}\right)\,\mathrm{d}x$

I am stuck on evaluating the following definite integral: $$\int_0^l (lx-x^2)\sin\left(\frac{n\pi x}{l}\right) \, \mathrm{d}x$$ Aprreciate any help!
4
votes
1answer
156 views

Integration of $\int^{1}_{-1} \frac {1}{3} \sinh^{-1} \left( \frac {3\sqrt 3}{2} (1-t^2) \right) dt$

Recently I came across with respect to this post of mine hyperbolic solution to the cubic equation for one real root given by $$ t=-2\sqrt \frac {p}{3} \sinh \left( \frac {1}{3} \sinh^{-1} \left( \...
4
votes
3answers
70 views

Integral of $\sin^5x\cdot\cos^{14}x$

I just don't understand what I'm doing wrong. I'm doing it exactly like it was taught to me but I'm getting a completely different answer from the correct answer. $$\int\limits_0^{\frac{\pi}{2}}\sin^...
1
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0answers
16 views

Finding force due to fluid pressure gives different answers

So there were two methods presented: the first one was using the $F = PhA$ where $P$ is density, $h$ is the height, $A$ is the area of the strip. The second one was using the center of mass which is $...
4
votes
1answer
80 views

Can we allow $f$ to be undefined at finitely many points in $(a,b)$ when formulating $\int_a^b f(x)\ dx=F(b)-F(a)$, ($F$ is the antiderivative of $f$)

Let $f$ be a real-valued function on a closed interval $[a,b]$ undefined only at finite points in $(a,b)$. Let $F$ be antiderivative of $f$. Then: $$\int_a^b f(x)\ dx=F(b)-F(a)$$ Is the theorem true?...
0
votes
1answer
41 views

Find leading approximation of oscillatory integrals (by integrating by parts)

I am thinking of integrals like the following $$\int_{0}^{1} e^{-x}e^{it(x^3-12x)}dx$$ where the critical point of $x^3 - 12x$ is out of the given interval $[0,1]$. How to find the leading ...
8
votes
1answer
136 views

Why does $\int_0^R 2 \pi r \,\mathrm d r$ give the area of a circle?

There's a method of computing the area of a circle by dividing it in concentric rings with infinitesimal width. Let $R$ be the radius of the circle and $r$ be the radius of the rings. The area of the ...
-1
votes
1answer
61 views

Find limit function of $\Phi(\alpha) = \int_\limits{0}^{\alpha}\ln(1+\alpha x)\,\mathrm{d}x$ [closed]

I have the integral: $$\Phi(\alpha) = \int_\limits{0}^{\alpha}\ln(1+\alpha x)\,\mathrm{d}x$$ I have to find a limit function but I have no idea where/how to start? I was thinking about L'Hopital ...
-1
votes
1answer
62 views

Evaluate $\lim_{x\to {\infty}} \frac{\int_1^x (t^2(e^{1/t}-1)-t)\,dt}{x^2\ln\left(1+\frac{1}{x}\right)}$

Calculate and evaluate the limit: $$\lim_{x\to {\infty}} \frac{\int_1^x (t^2(e^{1/t}-1)-t)\,dt}{x^2\ln\left(1+\frac{1}{x}\right)}$$ When plotting the upper and the lower part of the fraction ...
2
votes
1answer
34 views

Finding a difficult inverse Laplacetransform

I'm trying to solve the following problem: $$\mathcal{L}_\text{s}^{-1}\left[\frac{\text{F}(\text{s})+\text{G}(\text{s})}{1-\exp\left(-\frac{\text{s}}{4x}\right)}\right]_{\left(t\right)}\tag1$$ Where:...
0
votes
0answers
16 views

hemisphere pump problem

The tank (hemisphere) is full of water. Using the fact that the weight of water is $62.4\:\rm{lb/ft^3}$, find the work required to pump the water out of the outlet. The radius of the hemisphere is $10$...
1
vote
1answer
27 views

Calculate the derivative of underintegral function and find the integral

I have the integral: $$ \Phi(\alpha)=\int\limits_{1}^{2\alpha}\frac{\cos(2\alpha x^3)}{x}\,\mathrm{d}x \tag{1}$$ Have to find $\Phi'(\alpha)$ and calculate the integral then. do I understand ...
1
vote
1answer
63 views

$\int_0^\infty \sin{\left(\frac{1}{4x^2}\right)}\frac{\ln{x}}{x^2}dx$ [closed]

I want to solve the following integral but failed: $$\int_0^\infty \sin{\left(\frac{1}{4x^2}\right)}\frac{\ln{x}}{x^2}dx.$$ I wonder if you guys can help.
1
vote
0answers
39 views

Solving definite integral in two variables.

Solving a PDE with the following boundary problem with arbitrary constant $b$: $$u(0,t)=F(t)=b\int_0^\infty u(a,t)\mathrm{d}a$$ Hint given in the question is as follows: Split this integral in two ...