Questions about the evaluation of specific definite integrals.

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Consumer surplus for multiple products

I want to extend the classical definition of consumer surplus to multi-product case. For a single product, consumer surplus is given as \begin{equation} CS=\int_ {p_{market}} ^ {p _{\max}}D(p)dp. ...
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2answers
38 views

Limits and definition integrals involving logarithms

Let $a \in (0,1)$ and define $$I_n(a)=\int_a^1 (\ln x)^n \, \mathrm{d}x$$ Show that limit as $a\to 0$ we have, $$\lim_{a\to 0}I_n(a)=(-1)^n \cdot n!$$
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3answers
68 views

Definite integral with the squared cosine under the square root

I can't solve this $$\int_{0}^{5}{\sqrt{1+\left(\dfrac{\pi}{2}\cos(10 \pi x)\right)^2}dx}$$ My approach: If $10\pi x =u \to 10\pi dx=du$, so ...
2
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3answers
34 views

Can anyone prove the second property of a the following metric? $d: C([a,b])\times C([a,b]) \to R ; \ \ \ d(f,g)=\int_{a}^{b}|f(t)-g(t)|dt$ [on hold]

$$d: C([a,b])\times C([a,b]) \to R ; \ \ \ d(f,g)=\int_{a}^{b}|f(t)-g(t)|dt$$ $2.) d(f,g)=0 \iff f \equiv g$ Now in my notebook some lemma is called upon, concerning integrals, but it is unclearly ...
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1answer
29 views

Integrability of a function

Show that the function is integrable on $[0,2]$ $$f(x)=\left\{\begin{array}{cc} 1-x & x<1 \\ x^2-2x+1 & x \geq 1 \end{array}\right.$$ What conditions need to be checked in order for it to ...
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0answers
24 views

In the derivative of an integral, can I compare the size of the limit effect to the area effect?

I have minimal formal math training and sometimes encounter problems like this where I am not sure what relevant techniques are available to use. Thanks for any advice you can give. My integral looks ...
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0answers
29 views

Convergence of a sequence of integration

I am considering one problem and I am stuck in this step. The problem is that What conditions on function $f(u,\epsilon)$ are required to satisfy $$ \int_0^\epsilon f(u,\epsilon)\,du \rightarrow 0 ...
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1answer
22 views

Finding the surface area of the solid formed by a revolution of the function $f(y)=x$ when rotated about the line $y=0$.

I know of the following formulas for calculating surface areas: $\displaystyle A_S = 2\pi\int_{a}^{b}f(x)\sqrt{1+f'(x)^2}{\ dx}$ for the surface area ($A_S$) of the solid formed by revolving $f(x) = ...
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0answers
49 views

Can there be a unique function for $\int_{\frac{1}{n}}^n \sinh(x) \, dx$ when including parameters of $ 0 < n < 1 $ [on hold]

$$ \int_{\frac{1}{n}}^n \sinh(x) \, dx$$ For this function, depending on what the n value is, you end up getting different areas. For all values of $n$, the function creates: $ ...
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3answers
44 views

Solving the limit of an integral

Compute $$\lim _{n\to \infty }\left(\frac{\left(4n-4-2a_n\right)}{\pi }\right)^n$$ Where $$a_n=\int _1^n\:\frac{2x^2}{x^2+1}dx$$ The integral I solved and I got $a_n=2(x-\arctan(x))$ Afterwards, ...
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6answers
76 views

How to evaluate this integral $\int\limits_1^4\!\left( \frac{1}{\sqrt{x}}+\frac{1}{x}\right) \mathrm{d}x $?

$$\int_1^4\!\left( \frac{1}{\sqrt{x}}+\frac{1}{x}\right) \mathrm{d}x $$ The answer is $2+\ln(4)$, however I don't understand why. What I did was the following: $$\ln(x^{0.5})+\ln(x) = ...
2
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1answer
38 views

Help with Definite integral question

Anyone please help with this question: (a) Show that: \begin{align} \int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx \end{align} (b) Hence show that: \begin{align} \int_{0}^{\frac{\pi}{4}} ...
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0answers
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Find marginal distribution (Integral Solution)

I have derived bivariate exponential distribution in term of polar coordinate system. Now I need to derive marginal distribution of $f(\theta)$ from joint $f(r,\theta)$ for this we have to eliminate ...
2
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3answers
131 views

How do i evaluate this integral $ \int_{\pi /4}^{\pi /3}\frac{\sqrt{\tan x}}{\sin x}dx $?

Is there some one show me how do i evaluate this integral :$$ \int_{\pi /4}^{\pi /3}\frac{\sqrt{\tan x}}{\sin x}dx $$ Note :By mathematica,the result is : $\frac{Gamma\left(\frac1 ...
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2answers
24 views

What is the area of the part of the surface $z=yx$ bounded by $x^2+y^2=1$?

A parametrization of the part of the surface $z=yx$ bounded by $x^2+y^2=1$ is \begin{align} x &= u \cos v \\ y &= u \sin v \\ z &= \frac12 u^2 \sin 2v, \end{align} or $$r(u,v)=u \cos v \, ...
1
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1answer
13 views

Line integral of vector field/Why doesn't my solution work?

The question in its entirety: Determine for which constants A & B the vector field $$\mathbb{F} = (Axln(z))\mathbb{i} + (By^2z)\mathbb{j} + ((\frac{x^2}{z})+y^3)\mathbb{j}$$ is conservative. If ...
0
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1answer
12 views

Volumes of revolution (annular cross sections)

I have been asked to find the volume of the solid of revolution when the region bounded by the curve $y=x(4-x)$ and the $x$ axis is rotated about the $y$ axis. I am confused because the outer and ...
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0answers
43 views

Find the hydrostatic force using integration

A vertical dam has a semicircular gate. Find the hydrostatic force against the gate. The dam is 12 meters high, the water level is at 10 meters, and the semicircular gate had a diameter of 4 meters. ...
3
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1answer
151 views

Generalized Sophomore's dream. Question about originality

A few months ago I derived a beautiful fact: $$ \sum_{n=k+1}^\infty n^{k-n}=\int_{0}^{1} t^{k-t}dt~~~(*) $$ for every natural $k$. Generally: $$ \sum_{n=1}^\infty ...
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3answers
40 views

Convergence of improper integral?

Consider an improper integral such that: $$I = \int_0^{+\infty} \frac{f(x)}{x}dx.$$ If $\int_0^{+\infty}f(x)dx < + \infty$, Can we conclude that the integral I converges? Thanks for any answer or ...
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1answer
52 views

Finding a better upper bound for an integral of a product of $n$ terms

So I'm trying to find and upper bound for the integral $$ \int\limits_{a}^b \! (x-x_1)^2 \cdots (x-x_n)^2\, \mathrm{d}x, $$ where $x_i \in [a,b], \enspace \forall i=1,\dots ,n.$ I've tried ...
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1answer
45 views

Logarithmic Integral I

Consider the integral \begin{align} I = \int_{0}^{1} \frac{ \ln^{2}x}{(x^{2} - x + 1)^{2}} \, dx. \end{align} It is speculated that the value is \begin{align} I = \frac{10 \, \pi^{3}}{3^{5} \, ...
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2answers
32 views

(definite integral) area between two trig functions

I'm trying to figure out how to find the area between two trig functions. I know the procedure of integration here, finding the difference between two functions and integrating across whatever ...
2
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2answers
134 views

Evaluate the improper integral $\int_{0}^{\infty}{f(x)-f(2x)\over x}dx$, where $\lim_{x \to \infty} f(x) = L$ [duplicate]

Find $$\int_{0}^{\infty}{f(x)-f(2x)\over x}\, \mathrm{d}x$$ if $f\in C([0,\infty])$ and $\lim\limits_{x\to \infty}{f(x)=L}$. I tried denoting $\displaystyle \int{f(x)\over x}dx=F(x)$, but I don't ...
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0answers
26 views

Integral gaussian hypergeometric function

How can we define integral with interval $[b,\infty)$ $$ \begin{align} C(b,\alpha) & = \int_b^\infty \frac{1}{1+w^{\alpha/2}}\,\mathrm{d}w \\[8pt] & = 2\pi/\alpha \csc(2\pi/\alpha)-b_2 F_1 ...
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1answer
118 views

Definite integral with logarithm and arctangent inside of arctangent

How to prove $$\int_0^1 \left[ \frac{2}{\pi }\arctan \left(\frac 2 \pi \arctan \frac{1}{x} + \frac{1}{\pi }\ln \frac{1 + x}{1 - x}\right) - \frac{1}{2} \right]\frac{\mathrm{d}x} x = \frac{1}{2} \ln ...
6
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1answer
107 views

The quadratic and cubic versions of a tough intregral

In this post, Proving that $\int_0^1 \frac{\log \left(\frac{1}{t}\right) \log (t+2)}{t+1} \, dt=\frac{13}{24} \zeta (3)$, it's proved that $$I_1=\int_0^1 \frac{\log \left(\frac{1}{t}\right) \log ...
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0answers
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An integral involving the Gamma function

By utilizing the results of two previously asked questions, Series involving Laguerre polynomials and Integral of binomial coefficients, what is a resulting value of the integral \begin{align} ...
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0answers
13 views

Laplace Transform of Kelvin functions

What is the value of the Laplace transform, in terms of the G-function, \begin{align} \int_{0}^{\infty} e^{-st} \, t^{m} \, \left(ber_{\nu}^{2}(t) + bei_{\nu}^{2}(t)\right) \, dt \hspace{5mm} ? ...
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1answer
13 views

Integral identity for variable in integration limit

The following is an interesting integral identity: $H(t)=\int_0^tf(x,t)dx$, for $f(x,t)$ a sufficiently smooth function. Then, $H'(t)=f(t,t)+\int_0^tf_t(x,t)dx$. Why can't we use standard ...
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2answers
55 views

Convergence of $\int_0^\infty x^\alpha \cos e^x \, dx$

I tried to solve whether this integral is convergent or not and whether that convergence is conditional or absolute for a given $\alpha$. $$\int _0^{\infty }\:\:x^{\alpha \:}\cos\left(e^x\right)\, ...
16
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1answer
290 views

Proving that $\int_0^1 \frac{\log \left(\frac{1}{t}\right) \log (t+2)}{t+1} \, dt=\frac{13}{24} \zeta (3)$

Are we aware of an elementary way of proving that? $$\int_0^1 \frac{\log \left(\frac{1}{t}\right) \log (t+2)}{t+1} \, dt=\frac{13}{24} \zeta (3)$$ Of course, with the help of Mathematica it can be ...
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0answers
46 views

How to evaluate the integral ${\displaystyle\int_0^{m_1} \int_0^{m_2} }\frac{dx' dy'}{[(x-x')^2+(y-y')^2+25]^{\frac{3}{2}}} $

How to evaluate the integral $$\int_0^{m_1} \int_0^{m_2} \frac{dx' dy'}{[(x-x')^2+(y-y')^2+25]^{\frac{3}{2}}} $$
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1answer
30 views

Interchange Order of Integrals

Can someone explain the last step in this process. Specifically, how do you get the new limits of integration? Expected Value Definition: $E[Y] = \int_0^\infty{P\{Y \ge y\} \, dy}$ Expand: $E[Y] = ...
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1answer
43 views

Inequality using integrals and absolute values

Let $u,v$ be continous functions in $[a,b]$ a compact interval and let $c > 0$. Suppose that $\forall x\in [a,b]$, the following inequality is true: $$|u(x)-v(x)|\leq c\int^x_a|u(t)-v(t)|dt$$ ...
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1answer
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How to find $I=\int_{-4}^4\int_{-3}^3 \int_{-2}^2 \int_{-1}^1 \frac{x_1-x_2+x_3{-}x_4}{x_1+x_2+x_3+x_4} \, dx_1 \, dx_2 \, dx_3 \, dx_4$

How can I find this integral $$I=\int_{-4}^4\int_{-3}^3 \int_{-2}^2 \int_{-1}^1 \frac{x_1-x_2+x_3{-}x_4}{x_1+x_2+x_3+x_4} \, dx_1 \, dx_2 \, dx_3 \, dx_4$$
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2answers
48 views

Definite Integration, keep getting wrong answer.

Correct to 4 significant figures $$\int_{1}^{2}{\csc^24tdt}$$ Done this multiple times now and can't seem to get the answer at the back of the book. Here's my attempt: ...
2
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2answers
38 views

Why does this sum equal zero?

Le}t $\gamma$ be a piece-wise, smooth, closed curve. Let $[t_{j+1}, t_{j}]$ be an interval on the curve. Prove, $$\int_{\gamma} z^m dz=0$$ In the proof it states $$\int_{t_{j}}^{t_{j+1}} ...
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0answers
19 views

Showing certain sum as a Riemann-Stieltjes integral

Let $e(\beta) = e^{2 \pi i \beta}$. I am reading an article, where the author defines the following sum $$ S(N) = \sum_{0 \leq x \leq N, x \equiv g (mod \ q)} \Lambda(x) e(f(x) \alpha), $$ where $f$ ...
6
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3answers
83 views

How do I evaluate this : $\int_{0}^{\infty} \ln \left( 1 + \frac{a^{2}}{x^{2}}\right)\ dx $ for $a > 0$?

How do I evaluate this integral if I supposed that : $a > 0$ $$\int_{0}^{\infty} \ln \left( 1 + \frac{a^{2}}{x^{2}}\right)\ dx .$$ For $a=2$ I have got : $2\pi$ I think the result will be : ...
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1answer
25 views

Error bound of midpoint rules with unbounded second derivative

It is well known that error bound of midpoint rule for function $f[a,b]$ is given by $$ E\leq K\frac{(b-a)^3}{24 n^2} $$ where $|f(x)''\leq K|$ and $n$ is the number of time steps. if second ...
4
votes
1answer
68 views

Area under tangent to a curve.

The tangent to the graph of the function $y=f(x)$ at the point with abscissa $x=a$ forms with the line $x$-axis an angle $\frac{\pi}{6}$ and at the point with abscissa $x=b$ an angle of ...
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0answers
41 views

How to find the following definite and indefinite integrals [duplicate]

I want to calculate the integral $$\int_0^{\pi \over 2}e^{ \sin t}dt$$ can we find a primary function for $f(t) = e^{\sin t}.$ With many thanks.
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2answers
42 views

Finding the following definite and indefinite integrals

I want to calculate the integral $$\int_0^{\frac{\pi}{2}} e ^{ \sin t}\, dt.$$ Can we find a primitive function for $f(t) = e ^{\sin t}$?
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votes
2answers
62 views

Finding $\int_{0}^{\frac{\pi}{2}}\frac{1}{\cos (x-\frac{\pi}{3}).\cos (x-\frac{\pi}{6})}\mathrm{d}x$

How can I find $$\int_{0}^{\frac{\pi}{2}}\frac{1}{\cos (x-\frac{\pi}{3}).\cos (x-\frac{\pi}{6})}\mathrm{d}x$$ ? I suspect this has something simple to do with the basic definite integral properties; ...
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votes
2answers
49 views

How to calculate the integral?? [closed]

I want to calculate the integral $$\int_0^{\pi \over 2}e^{c \sin^2t}dt$$ for a constant parameter $c \in \Bbb R.$ With many thanks.
2
votes
1answer
59 views

Find the area bounded between $f(x)=\frac{\arctan(x)}{x^2}$ and $g(x)=\frac{\arctan(x)}{x^2+1}$

Find the area bounded between $$f(x)=\frac{\arctan(x)}{x^2} \quad\text{and}\quad g(x)=\frac{\arctan(x)}{x^2+1}.$$ The title says the question. The limits are from 1 to infinity. I know that I ...
6
votes
0answers
123 views

Integral of exponencial

Calculate the following integral $$\int_0^{+\infty} \exp\left(-a^2 x\left(\dfrac{x-6}{x-2}\right)^2\right) \dfrac{dx}{\sqrt{x}}$$ I think the relation between this integral and function gamma is ...
1
vote
1answer
17 views

Evaluating the volume of a torus formed by rotating a region about a horizontal axis using shells.

Using the method of cylindrical shells, find the volume of the shape created by revolving the region $x^2+(y-5)^2=4$ about $y=-1$. A cylindrical shell is given by: $2\pi v f(v) \ dv$ I solve ...
12
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1answer
207 views

Calculating 2 integrals in polylogarithmic functions

Are we aware of any nice way of calculating these $2$ integrals? $$i) \space \int_0^1 \frac{\text{Li}_2\left(x-x^2\right)}{x^2-x+1} \, dx$$ $$ii)\space \int_0^1 ...