Questions about the evaluation of specific definite integrals.

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Double integral over an annulus

Question: Let $D$ be part of the annulus $1\le x^2+y^2 \le 4$ lying in the first quarter of the $oxy$ plane where $x \ge 0, y \ge 0$ and below the line $y=x$ Evaluate the integral $$\iint_D\ ...
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11 views

Extending the Riemann integral to any compact set

One basically defines a Riemann integral on a closed interval. I'd like to extend the Riemann integral to any compact set. Let $K \in \Bbb R$ be compact. Let $f\colon K \rightarrow \Bbb R$, with the ...
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1answer
30 views

How to evaluate $\lim_{c \rightarrow \infty} \int_{-c}^c f(x)dx$

I'm trying to evaluate: $$\lim_{c \rightarrow \infty} \int_{-c}^c \frac{1+x}{1+x^2}dx$$ but I don't understand how to evaluate $$\lim_{c \rightarrow \infty} \int_{-c}^c f(x)dx$$ How?
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1answer
23 views

Evaluating a tricky exponential function integral

I am trying to evaluate the following integral $$ I = \int_0^t s^{2\alpha - 1} \exp\left(\frac{i \sqrt{2} \left(t^{2 \alpha + 1} - s^{2 \alpha + 1}\right)}{ 2 \alpha + 1}\right)\mbox{d}{s} $$ where ...
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6answers
155 views

What is $\lim_{p \to 0} \left(\int_0^1 (1+x)^p \, dx\right)^{1/p}$?

What is $$\lim_{p \to 0} \left(\int_0^1 (1+x)^p \, dx\right)^{1/p}\text{ ?}$$ I used binomial to get value as $2^p-1$ so limit becomes $$(2^p-1)^{1/p}.$$ But I can't go any further.
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3answers
59 views

Prove that $\int_{0}^{\frac{\pi}{2}}{\frac{\sin(2n+1)x}{\sin(x)}dx}=\frac{\pi}{2}$ for $n\ge0$

Prove that $$\int_{0}^{\frac{\pi}{2}}{\frac{\sin(2n+1)x}{\sin(x)}dx}=\frac{\pi}{2}$$ for $n\ge0$ I am not able to proceed with the integral. For the case $k+1$ please guide me through the problem. ...
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1answer
32 views

Calculate $\int_D \rvert x-y^2 \rvert dx \ dy $

$$\int_D \rvert x-y^2 \rvert dx \ dy $$ $D$ is the shape that is delimited from the lines: $$ y=x \\ y=0 \\ x=1 \\$$ $$D=\{ (x,y) \in \mathbb{R}^2: 0 \le x \le 1 \ , \ 0 \le y \le x \}$$ ...
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2answers
32 views

Definite integral of a positive continuous function equals zero?

Let's calculate $$\int_0^{\frac\pi 2} \frac {dx}{\sin^6x + \cos^6x}$$ We have $$\int \frac {dx}{\sin^6x + \cos^6x} = \int \frac {dx}{1 - \frac 34 \sin^2{2x}}$$ now we substitute $u = \tan 2x$, and get ...
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2answers
57 views

Areas under the graphs of $\frac{1}{x}$ and $\frac{1}{x^2}$ from $1$ to $\infty$

A simple evaluation of the definite integral tells us that the area under the graph of $[\frac{1}{x}]^2$ from $1$ to $\infty$ is finite whereas that of $\frac{1}{x}$ for the same limits is infinite. ...
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1answer
67 views

Brostein Integral 21.42

Good morning. I came across the following integral in some field theory calculation: $\int_0^\pi dx\,\log\left(a^2+b^2-2ab\cos x\right)=2\pi\log\left(\max\lbrace a,b\rbrace\right)$ for ...
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2answers
58 views

Which values $a,L$ satisfy $\frac{\int_0^{4π}e^{t}(\sin^{6}(at)+\cos^{4}(at))\,dt}{\int_0^{π}e^{t}(\sin^{6}(at)+\cos^{4}(at))\,dt}=L$?

Which option(s) below have the values of $a$ and $L$ that satisfy the following equation? $$\frac{\int_0^{4π}e^{t}(\sin^{6}(at)+\cos^{4}(at))\,dt}{\int_0^{π}e^{t}(\sin^{6}(at)+\cos^{4}(at))\,dt}=L$$ ...
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1answer
29 views

Help in evaluating an integral of exponential function

I am trying to evaluate the following integral $$ I = \int_{0}^{t}s^{-b-1}e^{-\frac{1}{2} a^2 s^{-2 b}} ds$$ where $a > 0$ and $ 0 \le b \le 1$. I am not quite sure how to solve this. Any help ...
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1answer
30 views

Drones and Integrals Project

Hello everyone and thanks for taking the time to read this post. So in my college calculus class we had the opportunity to fly a drone and get it's flight data. I have a spreadsheet featuring two ...
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1answer
116 views

Show that $\int_0^1\frac{1+x^8}{1+x^{10}}dx=\frac{\pi}{\phi^5-8}$

Prove that, $$ \int_0^1\frac{1+x^8}{1+x^{10}}dx=\frac{\pi}{\phi^5-8} $$ What kind of subsititution should be used to solve this integral Another integral that give the same answer but with a ...
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4answers
54 views

Find $\int_{-5}^{-3} \frac{dx}{\sqrt{x^2 - 4}}$ using trig substitution

$$\int_{-5}^{-3} \frac{dx}{\sqrt{x^2 - 4}}\\ x = 2\sec\theta \\ dx = 2\sec\theta \tan\theta \,d\theta\\ \theta \in[0,\frac{\pi}{2})\cup(\frac{\pi}{2},\pi]\\ $$ ...
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2answers
50 views

Find $\int_0^{\infty} \frac{dx}{1+e^x}$

$$\int_1^\infty\frac{dx}{1+e^x} $$ $$\lim_{M\to\infty}\int_1^M\frac{e^xdx}{e^x(1+e^x)} \\ u= 1 + e^x \\ du = e^x dx \\ \lim_{M\to\infty} \int_{1+e}^{1+e^M} \frac{du}{(u-1)u} $$ I then found the ...
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1answer
94 views

What's the integral of $\frac{1}{x^2}\csc^2\left(\frac{1}{x}\right)$?

It's known that $\int\csc^2(x)dx = -\cot(x) + C$, but I don't know how to integrate $\int\frac{1}{x^2}\csc^2(\frac{1}{x})dx$. Can you help? Answer to integral ...
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3answers
44 views

Find area of shaded area in curve with range of values for $y$

The parabola in the diagram has equation $y = 32 - 2x^2$ The shaded area lies between the lines $y=14$ and $y=24$ Looking at the graph, I only need to find half the area and multiply by ...
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1answer
72 views

Simple yet challenging integral, can it be solved analytically, and if so, the answer.

I'm trying to find solutions to the 3 following integrals. The first 2 are of the same form, only varying by a constant in the numerator within the cosine, and yes, x is a constant in the first one. ...
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2answers
56 views

find integral $\int_{1}^{-1} \sin\left(x^3\right) dx$

$$\int_{1}^{-1} \sin\left(x^3\right) dx$$ so I know the result is $0$ since above function is odd. But how to compute this integral?
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2answers
28 views

Given $\int _{-1}^{1}g(x)= 1$ show that $\int _{-1}^{1}f(x)g(x)\geq 1$ for certain $f,g$.

Let $f$ and $g$ be two positive valued functions defined on $[-1,1]$, such that $f(x)f(-x)=1$, and $g$ is an even function with $\int _{-1}^{1}g(x)= 1$. Show that $\int _{-1}^{1}f(x)g(x)\geq 1$. I ...
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1answer
37 views

Alternating Euler sums with even index

We are all aware of the generating function of $\frac{x \arctan x}{x^2+1}$ which is: $$\frac{x \arctan x}{x^2+1} = \sum_{m=1}^{\infty} (-1)^m \left ( \mathcal{H}_{2m} - \frac{1}{2} \mathcal{H}_m ...
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2answers
140 views

How to evaluate $\int_0^\infty \frac{e^{-x}+x-1}{x(e^{2x}-e^{-2x})}dx$? [on hold]

We are unable to verify this this equality $$ 4\int\limits_0^\infty \frac{e^{-x}+x-1}{x\left(e^{2x}-e^{-2x}\right)}\;\mathrm{d}x=\gamma+\ln\frac{16\pi^2}{\Gamma^4\left(\frac{1}{4}\right)}\;. $$ ...
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2answers
54 views

Evaluate the improper integral $\int_0^\infty {\exp(−sk)\over k}\sin(kx)\,dk$.

$$\int_0^\infty {\exp(−sk)\over k}\sin(kx)\,dk$$ I've tried hard for this but of no use.I've applied integration by parts by which I get $$\int_0^\infty \exp(-sk)\sin(kx)\,dk=\frac{x}{x^2+ s^2}.$$ ...
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1answer
19 views

Bessel Function Integral with sin argument

I would like to find if possible a solution (closed form) or approximation for the following integral: $$\int_{\pi/2}^{\pi}\int_{\pi/2}^{\pi}J_{0}\left(\alpha \sin\theta_{k}\right)J_{0}\left(\alpha ...
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1answer
16 views

Trouble with parametrizing function

hope you're all well! I just started learning about line integrals in class today, and I'm having a difficult time understanding how and why the solution manual came up with the parameterization for ...
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41 views

A variant of the exponential integral

Consider the following integral (for $x,y\in \mathbb{R}_{>0}$) $$E(x,y) = \int_0^1 \frac{\mathrm{e}^{-x/s-ys}}{s}\,\mathrm{d}s,$$ which is a variant of the usual exponential integral $E_1(x)$ to ...
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1answer
42 views

Evaluate Double Integral [on hold]

Evaluate the following double integral: $$\int _0^{2\pi }\int _0^1 \left(x - 2x^2 \sin\left(y\right) \cos\left(x^2+1\right)\right) \text dx\,\text dy$$ Please note that the answer is ${\pi}$
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2answers
79 views

Evaluating $\int_{0}^{3} \sqrt{1+x}\: dx$ using Limit of a Sum approach

Evaluate $\int_{0}^{3} \sqrt{1+x}\: dx$ using Limit of a Sum approach. Using the formula $$\int_{a}^{b} f(x)\:dx=(b-a) \times \lim_{n \to \infty} \frac{1}{n} \times ...
3
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1answer
98 views

Is there a closed form of this integral $ \int_0^\infty \sin(xe^{-x})dx\, $? [on hold]

I have tried by subsititution method and it got more complicate than before. Can anyone help me to evaluate this integral. $$ \int_0^\infty \sin(xe^{-x})dx\,. $$
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1answer
27 views

Why most of the books give Definite Integral Represents Area Under the curve

According to my understanding The definition of Definite Integral is: if $f(x)$ is a Continuous function in $[a \:\: b]$ and if $P$ is Partition of the Interval $[a \:\: b]$ Then $$ \lim _{\lVert P ...
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For which $a, b, $ and $c$ with $0 < a < b < c$ does $\int_0^1 f^a(x)dx =\int_0^1 f^b(x)dx =\int_0^1 f^c(x)dx $ essentially determine $f(x) $

For which values of $a, b, $ and $c$ with $0 < a < b < c$ does $\int_0^1 f^a(x)dx =\int_0^1 f^b(x)dx =\int_0^1 f^c(x)dx $ essentially determine $f(x) $. This is a generalization of Solve ...
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2answers
62 views

Does an analytical form exist for the following integral

I have an integral $$f(n,a)=\int_0^{2\pi}\mathop{\mathrm{d}x}\frac{\cos(nx)\cos^2x}{1-a\cos^2x},$$ where $n$ is an even integer and $0<a<1$ is a real number. Does an analytical form exist for ...
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Closed-form of an integral involving a Jacobi theta function, $ \int_0^{\infty} \frac{\theta_4^{10}\left(e^{-\pi x}\right)}{1+x^2} dx $

Motivation The Jacobi theta function $\theta_4$ is defined by $$\displaystyle \theta_4(q)=\sum_{n \in \mathbb{Z}} (-1)^n q^{n^2} \tag{1}$$ For this question, set $q=\large e^{-\pi x}$ and $\theta_4 ...
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1answer
40 views

The so-called error function defined as: $erf(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{-t^{2}}dt$

The so-called error function is defined as: $$erf(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{-t^{2}}dt$$ show that the function $y(x) = e^{x^2}erf(x)$ satisfies the differential equation: ...
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1answer
19 views

Dirac-Delta function representation - infinite sum involving trigonometric identities

Proof of the identity: $$\delta (x-x') = \sum_{n=0}^{\infty} \Big\{ \cos[n \pi(x-x')] - \cos[n \pi(x+x')] \Big\} $$ I can intuitively tell that this function is $\infty$ for $x=x'$, and zero ...
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1answer
30 views

Estimate $\int_0^{10} f(x)g'(x)dx$ if $f(x) = x^2$ and $g$ has the following values [closed]

I need help solving this problem... I believe it has something to do with either Riemann sums or integration by parts. Here is the problem. Estimate $\int_0^{10} f(x)g'(x)dx$ if $f(x) = x^2$ and $g$ ...
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1answer
45 views

$\iint y^2 dxdy$ over circular region

Suppose I want to calculate $$\iint y^2 dxdy$$ over region outside $$C_1=x^2 + y^2 = ax$$ and inside $$C_2=x^2 + y^2 = 2ax$$. How can we perform this integral? I approached this problem using polar ...
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2answers
647 views

Solve these functional equations: $\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$

Let $f : [0,1] \to \mathbb{R}$ be a continuous function such that $$\int_0^1\!{f(x)^2\, \mathrm{dx}}= \int_0^1\!{f(x)^3\, \mathrm{dx}}= \int_0^1\!{f(x)^4\, \mathrm{dx}}$$ Determine all such ...
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1answer
35 views

Simpson's rule is not producing better results than Riemann sums

I have to calculate RMS value $\sqrt {\int_0^T\frac 1T*f(t)^2dt} $ and I know from the maths that the Simpson's rule should provide better approximation of the definite integral than the Riemann sums. ...
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1answer
33 views

Sum of infinite circles on a plane

A power $P_0$ is spread on infinite circles of radius $r_0$ so the incident power on every circle is $dP_0$. The center of every circle is located on a plane $\Gamma =\left(0\le x\le x_0,0\le y\le ...
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1answer
18 views

How to find the inverse Fourier transfmation of exp(-sk)/k.

I've tried this with the help of hint given by one of my friend.He told me to first find the Inverse fourier transformation of exp(-sk) which is $$ \frac{\sqrt2}{\sqrt pi}\frac{x}{x^2+ s^2}$$ .After ...
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1answer
36 views

Area between two curves (Demidovich)

I'm trying to solve some problems on definite integrals from Demidovich's book and I'm stuck on calculating the area between two curves defined by: $$y_1 = \frac{a^3}{a^2 + x^2}, y_2 = 0$$ Any hints ...
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1answer
45 views

How to calculate a definite integral with complex numbers involved?

I'm trying to calculate this integral, and I find it difficult when coping with complex numbers. $$ f(k) = \int_{lnK}^{\infty} e^{ikx} (e^{x}-K) dx ...
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0answers
64 views

Limit involving integrals [closed]

Given $$I_n=\int_{0}^1\frac{x^n}{x+2014} dx$$ Solve $$\lim_{n\to\infty}nI_n$$
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1answer
49 views

Calculate this Triple integral!

They ask me find the following: W is the solid bounded by the limited right circular cylinder: $$ x^2+y^2=1$$ and the planes: $$z=0, z=4$$ must calculate: $$\iiint_W z\frac{e^{2x^2+2y^2}}{2} ...
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2answers
46 views

Finding an integral involving logarithmic functions: $\int_0^\infty\frac{1}{z[\ln(z)]^2}dz$ [closed]

Finding the following integral: $$\int_0^\infty \frac{1}{z[\ln(z)]^2} dz$$
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1answer
15 views

Boundary change in double integral: $\int_t^T \left( \int_t^u r(s)ds \right)du = \frac{1}{T}\int_t^T (T-s)r(s)ds$

I have following problem. I have been reading an article on pricing Asian options and I have found one article directly concerning my topic. However, it is horribly written and I am trying to ...
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1answer
30 views

Finite Integral involving Bessel Function, $J_1$

Is there a closed form solution for the following integral $$ \int_0^a dx \ x \ J_1(x) \ , $$ where $J_1(x)$ is a Bessel Function and $a>0$ is a finite real number? Thanks.
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5answers
92 views

Evaluate $\int_0^5 e^{-2t}\sin(t)\, \mathrm{d}t$.

$$\int_0^5 e^{-2t}\sin(t) \,\mathrm{d}t$$ I know I should be able to integrate this by parts but I can't seem to get the parts I choose to make the result any easier to integrate.