Questions about the evaluation of specific definite integrals.

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$\int^\infty_0 e^{-\alpha x}\sin(\beta x)\,dx = \frac{B}{\alpha^2+\beta^2}$ Laplace

$$ \int^\infty_0 \! e^{-\alpha x} \sin(\beta x)\,dx = \frac{B}{\alpha^2+\beta^2} $$ Can someone start this for me? I don't know where to begin.
3
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0answers
20 views

Solution of Integral $\int_0^\infty \frac{x^{r/\beta}e^{-\alpha x}}{\left(1-(1-\beta)e^{-x}\right)^{\alpha+1}} \, dx$

I need to solve the following integral $$\int_0^\infty \frac{x^{r/\beta}e^{-\alpha x}}{\left(1-(1-\beta)e^{-x}\right)^{\alpha+1}} \, dx$$ where $r,\alpha,\beta>0$. kindly help me to solve this ...
2
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1answer
33 views

positive double integrals

Suppose $U:={\iint}_{R} (x^2 + 2y^2+9) \,dx\,dy$ and $V:= \iint _R (2x^2 + 3y^2)\, dx\,dy$. Determine the integration region $R$ where $U \geq V$. Hence, find the value $U-V$ over this region. ...
2
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1answer
74 views

Bonus integration problem we got at class: Integrate $\frac {x \sin x}{1+\cos^2x}$ between $0$ and $\pi$ [duplicate]

Bonus integration problem we got at class: Integrate $\frac {x \sin x}{1+\cos^2x}$ between $0$ and $\pi$ So the lecturer gave this problem. I tried this really hard but couldn't succeed. It ...
0
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2answers
74 views

Why $\int _c^df^{-1}\left(y\right)\:dy+\int _a^b\:f\left(x\right)dx=b\cdot d-a\cdot c$? [duplicate]

Why $\int _c^df^{-1}\left(y\right)\:dy+\int _a^b\:f\left(x\right)dx=b\cdot d-a\cdot c$ ? where f is an bijective function and $f(a)=b,f(c)=d,$ I don't understand graph... I can't see on graph this ...
1
vote
1answer
32 views

It's possible to prove that this integral is positive?

I would like to prove that this function $$F_t(x)=\int_{-\infty}^{\infty}e^{-\frac{u^2}{2 t}}\,\cosh\left(u\right)\,K_{1/2+i\,u}\left(\frac{1}{4x}\right)\,du,\hspace{0.5cm}x,t>0$$ is positive, ...
0
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2answers
35 views

How to fastest approximate definite integrals

I know that a definite integral is a limit of Riemann sums. So if one wanted to estimate a definite integral (because one might not be able to find an antiderivative), then one can just take enough ...
0
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1answer
24 views

How do you integrate $\int _{0}^t\:\dot p(s) p(s) + p^2(s)ds$

Given $p(s)$ some single valued function How can I show that $$\int _{0}^t\:\dot p(s) p(s) + p^2(s)ds$$ has resulting in something along the line of $$\frac{p^2(s)}{2}$$ note $\dot p(s)$ signifies ...
3
votes
3answers
228 views

Integral involving Bessel functions of the first kind

I am stuck with the following integral. Does it converge? $$ \int_{0}^{\infty}\left(J_1(x)^2+J_1(x)J_1(x)^{''}\right)\text{d}x $$ According to tables I find that the first term is divergent, so I ...
0
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3answers
54 views

How to find $p(t)$ when $m$ varies linearly with $t$? [on hold]

I have a function $p(t)$ (position and time) defined by $$p(t) = \frac{1}{2} \cdot \frac{F}{m} \cdot t^2$$ when the mass is constant. This is derived from Newtons second law and by integration of the ...
2
votes
4answers
102 views

Compute $\int _{\frac{4}{5}}^2\:f^{-1}\left(x\right)dx$

We have to compute $\int _{\frac{4}{5}}^2\:f^{-1}\left(x\right)dx$ where $f\left(x\right)=\frac{-x^3+2x^2-5x+8}{x^2+4},\:x\in \mathbb{R}$ is an bijective function. How help if we kno![enter image ...
0
votes
2answers
41 views

Help me with Integral, $\int_1^{\sqrt 3}\frac{dx}{(1+x^2)\arctan x}$ [on hold]

$$\int_1^{\sqrt 3}\frac{dx}{(1+x^2)\arctan x}$$ Can you help me with this integral?
1
vote
1answer
26 views

The value of x satisfying $\int^{2[x+14]}_0\{\frac{x}{2}\}dx =\int^{\{x\}}_0[x+14]dx $ where [.] …

Problem : The value of x satisfying $\int^{2[x+14]}_0\{\frac{x}{2}\}dx =\int^{\{x\}}_0[x+14]dx $ where [.] denotes the greatest integer function and $\{.\}$ denotes the fractional part function. ...
1
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0answers
38 views

Is there a closed form expression for the following definite integrals?

I am looking for a closed form for these two integrals $$\int_{-\infty}^{-a}\text{d}x \frac{1}{|x|}e^{-\frac{1}{2}x^2\sigma^2}e^{i k |x|}+\int_a^{\infty}\text{d}x ...
0
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1answer
41 views

Evaluate $\lim _{x\to \infty }\int _{\frac{1}{x}}^x\:f\left(t\right)dt$ and a big mistake in the book

We have to evaluate $\lim _{x\to \infty }\int _{\frac{1}{x}}^x\:f\left(t\right)dt$ where $f\left(t\right)=\frac{1}{\left(1+t^2\right)\left(1+t^3\right)}$. In my book they say that $\int ...
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2answers
43 views

Is this integral correctly calculated?

The problem is that I can't use wolframalpha to check this because he is worried about integration limits: I have $a>0$ and $t \in (-1,1).$ $$(1-t)^{\frac{a}{2}} \int_0^t \frac{1}{(1-x)^{a+1} }dx= ...
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1answer
27 views

How to show this integral (Error function)

I'm given this question. Show that $$\int_{0}^{0.25}\frac{1}{\sqrt{x}}e^{-x}dx=\int_{0}^{0.5}2e^{-u^2}du$$. As I know this integral is an error function. How to show? Can anyone give me some hints? ...
7
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5answers
167 views

Evaluate $\int_0^{1/\sqrt{3}}\sqrt{x+\sqrt{x^2+1}}\,dx$

I want to find a quick way of evaluating $$\int_0^{1/\sqrt{3}}\sqrt{x+\sqrt{x^2+1}}\,dx$$ This problem appeared on the qualifying round of MIT's 2014 Integration Bee, which leads me to think ...
0
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2answers
81 views

Evaluate $\int_{-\pi}^\pi \! \cos(kx)\cos^n(x) \, \mathrm{d}x$

My question is: Evaluate $$\int_{-\pi}^\pi \! \cos(kx)\cos^n(x) \, \mathrm{d}x$$ for $k=0,1,...,(n-1)$ and $n \in \mathbb{N}$. I've tried integration by parts but without much success. Any ...
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2answers
23 views

How to find volume of the given solid analytically?

Here is the question - I am able to visualize the solid, but how do I find its volume? I'm unable to figure out the 2D structure that when rotated, produces this solid. Please help. Edit: The ...
2
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1answer
70 views

What is $\int_0^1 \frac{\log(x+1)}{x^2+1}dx$ [duplicate]

It's so deceptively simple and none of the usual techniques are working. Any and all insights are welcome.
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0answers
46 views

Estimation of a certain Integral

I estimated (w.r.t. $\varepsilon$) the expression \begin{align} &\left|\int_{-1}^{x_0-\varepsilon} (1-x)^{n-p}(1+x)^p+\int_{x_0+\varepsilon}^1 (1-x)^{n-p}(1+x)^p \, dx \right | \\[6pt] \leqslant ...
0
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1answer
30 views

The Gherkin (an egg shaped building) - equation for the curve in order to calculate the surface area of revolution

I am trying to calculate the surface area of revolution for The Gherkin, an egg-shaped building in London, UK. Not sure about how to obtain the equation of the curve but I have the data points that ...
1
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2answers
53 views

How we can prove that: $\sum _{k=n}^n f\left(\frac{k}{n}\right)\le n\cdot \log(2)$?

$f:\left[0,1\right]\rightarrow R,\:f(x)=\frac{1}{1+x}$ and we have to show that $\sum_{k=n}^n f\left(\frac{k}{n}\right)\le n\cdot\log(2)$. What I know is just that: $n\cdot \log(2)=\int_0^1 ...
2
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1answer
31 views

An integral related to the derivative of Legendre polynomials

I want to calculate the integral $$ I=\int_{-1}^{1} \Big(\frac{\mathrm{d}P_{n+1}(t)}{\mathrm{d}t}\Big) \Big(\frac{\mathrm{d}P_{m+1}(t)}{\mathrm{d}t}\Big) \mathrm{d}t $$ where $P_n(t)$ is Legendre ...
1
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1answer
64 views

Definite integral involving 2015

Evaluate $$\displaystyle\int_{2}^{2014} \frac{\log \left( 2015 - x\right )}{\log \left( 2015 - x\right ) + \log \left( x - 1\right )} \mathrm{d}x$$ I got the solution using software, and it is a ...
0
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0answers
49 views

how to solve this integral involving any square root

how to solve the integral $\int\sqrt{\alpha+\beta e^{\gamma t}}dt$ i got this integral from the problem Given that the velocity $v$ of a body $t$ segonds after passing a point $O$ is found by ...
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3answers
86 views

Integrating $f(x) = 1/x$ from $x=a$ to $x=\infty$

Can the integration of $f(x)=1/x$ from $x=a > 0 $ to $x=\infty$ ever be finite? That is, can $\int_{x=a}^{\infty} 1/x$ be finite?
2
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0answers
43 views

Computing an integral using residues

I am trying to find an integral: $$\int_{-\infty}^{+\infty}\frac{e^{-\sqrt{(x^2 + 1)}}}{(x^2 + 1)^2}\,\mathrm dx$$ I went about applying contour integral over a semicircle with diameter along $ x = ...
4
votes
3answers
112 views

Compute $\int_{0}^{\infty}\frac{x \log(x)}{(1+x^2)^2}dx$

Given $$\int_{0}^{\infty}\frac{x \log(x)}{(1+x^2)^2}dx$$ I couldn't evaluate this integral. My only idea here was evaluating this as integration by parts. \begin{align} \int\frac{x ...
2
votes
1answer
52 views

Where am I wrong in the following problem?

We have: $f:R\rightarrow R,\:\:f\left(t\right)=At^2-2Bt+C,\:where\:A=\int _1^2\:\frac{1}{x^2}dx,\:B=\int _1^2\:\frac{e^x}{x}dx,\:C=\int _1^2\:e^{2x}dx$ and we need to show that ...
4
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2answers
107 views

Evaluate $\lim_{n \to \infty} \int_{0}^1 \frac{n+1}{2^{n+1}} \left(\frac{(t+1)^{n+1}-(1-t)^{n+1}}{t}\right) \mathrm{d}t$

Evaluate $$\lim_{n \to \infty} \int_{0}^1 \frac{n+1}{2^{n+1}} \left(\frac{(t+1)^{n+1}-(1-t)^{n+1}}{t}\right) \mathrm{d}t$$ For this integral, I have tried using integration by parts and then ...
0
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1answer
26 views

Divergence of $\iint \text e^{-(\tau_1-\tau_2)}\,\theta(\tau_1-\tau_2)\,\text d ^2\tau$

Does this integral ($\alpha>0$) $$ I=\int_{-\infty}^\infty\text d \tau_1 \int_{-\infty}^\infty\text d \tau_2 \; \text e^{-\alpha(\tau_1-\tau_2)}\theta(\tau_1-\tau_2) $$ diverge? Here $\theta$ is ...
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2answers
37 views

Misunderstanding inequalities of integrals

We have to prove the following inequalities: 1) to show that $\frac{2x}{\pi }<sin\left(x\right)<x,\:and\:after\:1-e^{-\frac{\pi }{2}}\le \int _0^{\frac{\pi ...
2
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2answers
26 views

A question involving finding values in integrals?

Let p(x) be a continuous function such that $\int_2^3{p(x)}dx$=$c\cdot\int_0^2{p(\frac{x+4}{2}})dx$ then find the value of c? I am thinking of dividing the integral on the left hand side into two ...
0
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0answers
6 views

Hydrostatic Force on a submerged plane.

I am having trouble with question 5, it reads (ignore the Riemann sum part) : Here is what I did, and where did I go wrong?. The answer the book gives is: $6.7\cdot 10^4N$. Thank you.
1
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1answer
54 views

Evaluate $\int_{0}^{1} \frac {\ln x}{1-x^2} \mathrm{d}x $

I found this question in a reference book: $$\int_{0}^{1} \frac {\ln x}{1-x^2} \mathrm{d}x $$ Can Anyone give me Idea how do I begin solving this?
1
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2answers
27 views

sum approximation of a Lipschitz-continuous function

Let $f: [0, 1] \to \mathbb{R}$ be a Lipschitz continuous function with a Lipschitz constant $L > 0$, meaning: $$|f(x) - f(y)| ≤ L|x - y| \space\space\space \forall x, y \in [0, 1]$$ For the ...
0
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3answers
63 views

Integrate problem

We have to integrate $\int _0^{\pi }\:\left|\sin\left(2x\right)\right|dx$ and in my book, they split integral: $\int _0^{\pi }\:\left|\sin\left(2x\right)\right|dx=\int _0^{\frac{\pi ...
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0answers
40 views

Calculate an integral with delta function

In order to calculate the integral $$ f(x) = \frac{2}{\pi}\int_0^{\pi/2}\delta\Big(x-\sqrt{1\pm\sqrt{1-\beta^2\sin^2t}}\Big)\mathrm{d}t $$ where $\beta\in(0,1]$. I am hunting for a better solution, ...
1
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3answers
51 views

Integral of a tangent function

$$ \displaystyle {\int_{0}^{z}} \sqrt {1 + \tan^2(\dfrac{\pi}{4} \dfrac{z}{H} )} dz $$ _ $$ gives $$ _ $$ \dfrac{4H}{\pi} {\sinh^{-1}} ( {\tan \dfrac{\pi}{4} \dfrac{z}{H} } ) $$ Please advise ...
3
votes
1answer
48 views

How we can show that $\:I_n\ge \frac{2}{\pi }\left(\frac{1}{n+1}+\frac{1}{n+2}+…+\frac{1}{2n}\right)$

We have $I_n=\int _{\pi }^{2\pi }\:\frac{\left|sin\left(nx\right)\right|}{x}\:dx,$ and we need to show that$\:I_n\ge \frac{2}{\pi }\left(\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n}\right)$ I write ...
0
votes
2answers
88 views

Integration of $x^n e^{-x} dx$

I've been trying solve this, and even though I feel I'm really close to the answer- I'm quite unsure of the actual answer. The question is a definite integral $$\int_{0}^{\infty} \frac {x^n} ...
0
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0answers
6 views

Setting up triple integral in spherical coordinates

Integrate $f(x, y, z) = z^2$ over $A = \{(x, y, z) \in \mathbb{R^3} | x^2+y^2+z^2 \leq R^2, x^2 + y^2 + z^2 \leq 2RZ\}$ I know $A$ is the intersection between two spheres but I am unable to figure ...
0
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1answer
50 views

How to solve this integral in moment generating function

The moment generating function of generalised Pareto distribution eventually comes down to the following integral (here). $$ M_X(\theta) = \mathbb Ee^{X\theta} = \int_\mu^\infty e^{\theta ...
0
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0answers
13 views

Fundamental Theorem of Calculus for Line Integrals

Use the Fundamental Theorem of Calculus for Line Integrals to compute $\int_C F*dr$ where $$F(x,y,z)=(yz+2x)i+(xz+2y)j+(xy-2z)k$$ and C is the path from $(1,6,-1)$ to $(5,2,3)$ given by $x(t)=2t+1, ...
-2
votes
2answers
57 views

$\int _{k\pi }^{\left(k+1\right)\pi }\:\left|\sin\left(x\right)\right|dx$ [closed]

How can I solve the following integral? $\int _{k\pi }^{\left(k+1\right)\pi }\:\left|\sin\left(x\right)\right|dx$
5
votes
1answer
84 views

Can $\oint_{|z|=2}z^3 \bar {z} e^\frac{1}{(z-1)} dz$ be solved?

How we can calculate the result of following Integral? $$\oint_{|z|=2}z^3 \bar {z} e^\frac{1}{z-1} \mathrm{d}z$$
0
votes
1answer
20 views

Leibniz rule for an improper integral

It follows from leibniz rule that if $\frac{\partial f}{\partial \theta_0}(\theta,\theta_0)$ exists then $$\frac{d}{d\theta_0}\bigg(\int_0^{\theta_0}f(\theta,\theta_0)d\theta\bigg)=\int ...
1
vote
2answers
87 views

I can't understand how to prove this inequality

I don't understand how we can prove that inequality, without integration $$\frac{1}{x}\int_x^{2x} \left(2-\frac{1}{y+2}\right)\,dy \geq 2 - \frac{1}{x+2}.$$ P.S: Here is what I try... if can someone ...