Questions about the evaluation of specific definite integrals.

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order of integrals with independent limits

I was wondering if the following is true assuming that the limits are independent (like constants) $$ \int_{\alpha}^{\beta} \int_{\gamma}^{\psi} {xy} dx dy = \int_{\gamma}^{\psi} ...
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1answer
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Error estimate of definite integral of a taylor expanded function

If I consider a monotonic decreasing function $f(x)$ in the interval $[0,+\infty[$, and I consider the definite integral $\int_{0}^{+\infty}f(x)\,\mathrm{d}x$. What is the error committed if I compute ...
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Motivation behind parameters

This article shows a technique of evaluating a definite integral by introducing a suitable parameter. This however doesn't throw light on motivation for introducing that particular parameter. For ...
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0answers
31 views

Find $\int \tan(\tan x)\hspace{1mm}dx$

Find $\int \tan(\tan x)\hspace{1mm}dx$ This is an Interesting problem, which I have been trying from different directions, nothing seems to work, its been a day on this one. Can anyone figure out ...
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1answer
28 views

Finding area between two cosine curves

I must to find the area between these two curves: $$y = 2 \cos 7x, y = 2 − 2 \cos 7x$$ $$0 ≤ x ≤ π/7$$ And this is all I have so far: $$ 2\cos7x=2-2\cos7x $$ $$4\cos7x=2$$ $$\cos7x=1/2$$
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Closed form for ${\large\int}_0^1\frac{\ln^{\color{magenta}3}x}{\sqrt{x^2-x+1}}dx$

This is a follow-up to my earlier question Closed form for ${\large\int}_0^1\frac{\ln^2x}{\sqrt{1-x+x^2}}dx$. Is there a closed form for this integral? ...
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1answer
43 views

How to simplify this complex integral? [on hold]

How to approximate this integral as a function of a and b? $$\int_0^\pi\int_0^{2\pi}\sqrt{(a-b\sin\varphi\cos\theta)^2+(b\cos\varphi)^2+(b\sin\varphi\sin\theta)^2}d\theta d\varphi$$ where a and b ...
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4answers
216 views

Finding the definite integral of a function that contains an absolute value

The integral in question is this: $\int_{-2\pi}^{2\pi}xe^{-|x|}$ My attempt: Since there is a modulus, we split it up into cases. I'm not really sure which cases to split it into, do I just ...
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0answers
28 views

Could someone help find the shell height?

I am trying to solve this problem and have been going at it for 3 hours and not getting anywhere. I think I am suppose to have everything in terms of y but the x equals functions are throwing me off. ...
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1answer
35 views

Integrals involving roots

I am bit stucked with an integration form while doing one of my proofs for a graphics application.Issue is I cant take out the terms from the trigonometric functions for a proper known integral ...
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2answers
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Find $\int t\sin^{-1}t\hspace{1mm}dt$

Find $\int t\sin^{-1}t\hspace{1mm}dt$ How do we approach this question, is there a simple way to integrate
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1answer
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Integral from 0 to 16 of $\sqrt{x}/(x-4)$?

$$\int_{0}^{16}\frac{\sqrt{x}}{x-4}dx$$ So I'm letting $u=\sqrt{x}$, $du=1/2\sqrt{x}$, $u^2=x$ and $dx=2\sqrt{x}du$. I just don't really know what to do from here. I am trying different things and ...
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3answers
40 views

Using the comparison test to evaluate $\int_1^\infty\frac{1}{1+x^2+16x^4}dx$?

So using the comparison test to evaluate $\int_1^\infty\frac{1}{1+x^2+16x^4}dx$, and we're given $\int_1^\infty\frac{1}{4x^2}dx$. So I have been trying to set up an inequality to use, but I can't seem ...
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2answers
18 views

Calculate the Area of the space defined by two lines $\varepsilon_{1},\varepsilon_{2}$ and a curve $c_{1}$

I'm starting a class on Advanced Mathematics I next semester and I found a sheet of the class'es 2012 final exams, so I'm slowly trying to solve the exercises in it or find the general layout. I will ...
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4answers
82 views

Does the following integral converge: $\int_6^{\infty}\frac{dx}{\sqrt{1+x^2}}$

Does the following integral converge: $$\int_6^{\infty}\frac{dx}{\sqrt{1+x^2}}$$ I suppose we have to solve such problems by comparison test. All the integrals I tried so far do not fit the ...
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0answers
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Integral $\int_0^1 \frac{\sqrt{1-x}}{\sqrt{1+x^2}} dx$

Looking for a closed-form of this integral. $$I=\int_0^1 \frac{\sqrt{1-x}}{\sqrt{1+x^2}} dx$$ I'm looking for a closed-form of $I$ without using Meijer G-function, elliptic integrals or generalized ...
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1answer
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How can I solve this integral analytically or numerically

Hi I have an integral to do $$\nu =\int_{0}^{P(r)} \,\frac{dP}{P+\beta\rho(P)}$$ here I calculated $$\rho = 0.003 P^{\frac{2}{4}}+ 0.002P^{\frac{2.5}{4}}+0.0019P^{\frac{3}{4}}$$ My question can ...
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3answers
30 views

Finding the integral of $\int_{-\infty}^{\infty}e^{-|4x|}$.

So I am trying to find the integral of $\int_{-\infty}^{\infty}e^{-|4x|}$. I know the integral converges, and I know the answer as well, but I am confused on how to get the correct answer. My problem ...
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2answers
45 views

Find $\int_0^{\pi}\sin^2x\cos^4x\hspace{1mm}dx$

Find $\int_0^{\pi}\sin^2x\cos^4x\hspace{1mm}dx$ $ $ This appears to be an easy problem, but it is consuming a lot of time, I am wondering if an easy way is possible. WHAT I DID : Wrote this as ...
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2answers
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+300

Closed-forms for several tough integrals

These integrals came up in the process of finding solution to Vladimir Reshetnikov's problem. I wonder if there are closed-forms for the following integrals: \begin{array}{1,1} &[\text{1}] ...
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0answers
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Mellin transform with compact support

Mellin transform for $f(x)$ is usually defined as: $$F(s)=\int_0^\infty f(x)x^{s-1}dx$$ Is there a Mellin transform with compact support? For example like $$F(s,a,b)=\int_a^{b} ...
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+300

Integration of product of functions

Sir, I have been doing a proof related to one research topic. But after a long effort, I got ended up in a messy integration equation. Could you give me some suggestions to solve this equations? (Any ...
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6answers
187 views

How to show this integral equals $\pi^2$?

I was trying to evaluate an integral related to the product of two cauchy distributions and in one of the steps got stuck in the integral $$\int_0^{\infty} \frac{\ln(x)}{\sqrt{x}(x-1)} dx. $$ I ...
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2answers
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Integral with rational functions of powers and exponentials

Any ideas how to solve: \begin{equation} \int_0^\infty x^{n+\frac{1}{2}} (e^{a x }-1)^{-\frac{1}{2}} e^{i x t} dx \end{equation} where $a$ and $t$ are real, positive constants; $n$ is a positive ...
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4answers
150 views

Find $\int_0^{1/2} \sqrt{1+\sqrt{1-x^2}}\hspace{1mm}dx$ [on hold]

Find $\int_0^{1/2} \sqrt{1+\sqrt{1-x^2}}\hspace{1mm}dx$ How do we approach this problem, can someone explain
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Solving complex integral

I have the next integral: $$\int_{-\infty }^{0}\left |e^{-iw_1 x}+\frac{w_1-w_2}{w_1+w_2}e^{iw_1 x} \right |^2\mathrm{d}x$$ where $w_1$ and $w_2$ are real constants. After some algebraic process: ...
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3answers
244 views

Integral $\int_{0}^1\frac{\ln\frac{3+x}{3-x}}{\sqrt{x(1-x)}}dx$

I have a problem with the following integral: $$ \int_{0}^{1}\ln\left(\,3 + x \over 3 - x\,\right)\, {{\rm d}x \over \,\sqrt{\,x\left(\,1 - x\,\right)\,}\,} $$ The first idea was to use the ...
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2answers
231 views
+150

Integral of Combination Log and Inverse Trig Function

Does the following integral have a closed-form \begin{equation}\int_0^1\frac{\ln x}{1+x}\arccos(x)\,dx\,?\end{equation} This integral has been posted in Integral and Series a week ago but it ...
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4answers
107 views

Integral $\int_{1}^{\infty} \frac{\log^3 x}{x(x-1)} dx$

How do I arrive at the closed form expression of the integral $$\displaystyle\int_{1}^{\infty} \dfrac{\log^3 x}{x(x-1)}dx$$ Most probably the closed form is $\dfrac{\pi^4}{15}$
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3answers
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General closed form of an integral

I once asked a question about how to integrate the reciprocal of the square root of cosine. Is there a general closed form for the integral $$\int_{0}^{\theta_o} \dfrac{1}{\sqrt{\cos \theta-\cos ...
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1answer
25 views

Multiple integral 3 dimension

Find the volume of the body $$ v:{(x,y,z) :\quad x^2+y^2\le z \le \sqrt{2-x^2-y^2}}.$$ I really don't know what to beside that i have to do triple integral of one. My main problem is to ...
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4answers
124 views

Calculus Question: Improper integral $\int_{0}^{\infty}\frac{\cos(2x+1)}{\sqrt[3]{x}}dx$

How to evaluate integral $$\int_{0}^{\infty}\frac{\cos(2x+1)}{\sqrt[3]{x}}dx?$$ I tried substitution $x=u^3$ and I got $3\displaystyle\int_{0}^{\infty}u \cos(2u^3+1)du$. After that I tried to use ...
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0answers
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An integral with a decaying exponential with rational exponent

I was working on some mathematical derivations while I faced this integral: $$\Large \int_0^\infty x^{\alpha-1}e^{-\beta x} e^{-\lambda \left[\frac{x^2}{2x+\eta}\right]}\ \mathrm{d}x \quad .$$ Does ...
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3answers
85 views

Find $\int \sinh^{-1}x\hspace{1mm}dx$

Find $\int \sinh^{-1}x\hspace{1mm}dx$ $ $ I am asked to use the following Equation: $$\int \tan^{-1}x\hspace{1mm}dx= x\tan^{-1}x-\ln(\sec(\tan^{-1}x))+C$$ $ $ The confusing part is : What has ...
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0answers
43 views

How do I solve this tricky definite integral ?! [duplicate]

$$I=\int\limits_0^1\dfrac{x^2-1}{\ln x}\mathrm dx$$ I tried numerous substitutions but nothing seems to work.. any ideas ???!
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1answer
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Calculate the expected value

To get the expected value of $E(X), E(Y) $ and $E(X, Y)$ given: $$ f_{X,Y}(x,y) = 3x $$ where $0\le x \le y \le 1.$ My solution is, first get the margin distribution: \begin{aligned} f_x(x) &= ...
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An integral similar to integrating the Beta function

Peace be upon you, I have the following definite integral for the mathematical expectation of some distribution \begin{align*} \int_{-1}^1 z^2\int_{\max(z-1,-1)}^{\min(0,z)} (z-y)^a(-y)^b \ ,dy \,dz ...
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0answers
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How to evaluate the integral $e^{-(c\ln(\frac{1}{x}))^s} dx$?

Can anyone help me evaluate $$\int_{\alpha}^1 \exp{\left\{-\left(c\ln\left(\frac{1}{x}\right)\right)^s\right\}} dx$$, Where $0 \leq \alpha \leq 1$ and $s \in \mathbb{R}$. I tried changing ...
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1answer
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Integral evaluation with exponentials

I want to evaluate the integral $\int_0^T e^{-ax}e^{-bx^2} \, dx$. I found a direct solution: $$\int_{0}^{\infty} e^{-ax}e^{-bx^2} \, dx = \sqrt\frac{\pi}{b} \exp\left(\frac{a^2}{4b}\right) ...
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Evaluating $\;\int_{1}^{\ln3}\frac{e^x - e^{2x}}{(1 + e^x)}\,dx$

Find $\int_{1}^{\ln3}(e^x - e^{2x})/(1 + e^x)dx$. I looked through my notes for integration techniques and thought I could try a $u$ substitution but whatever I set $u$ to I can't seem to ...
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3answers
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Compute variance, using explicit PDF

I'm trying to get $\text{Var}(x)$ of $f(x) = 2(1+x)^{-3},\ x>0$. Please tell me if my working is correct and/or whether there is a better method I can use to get this more easily. $$ ...
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2answers
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Help finding k. Issue with integration

Let the continuous random variable $X$ have a probability density function $f(x)$ such that $$f(x) = k(1+x)^{-3}, x>0$$ $=0$ elsewhere Find k This is what I tried: $\int_0^\infty k(1+x)^{-3}dx ...
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How do I do this [closed]

I want the steps (and possibly proof for any answer you provide) to solve this problem!
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1answer
39 views

Are all definite integrals considered functionals?

In my Optimization class, we are messing around with some Calculus of Variations in an effort to find functions which minimize functionals. In these cases, the spaces we're working with are spaces ...
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Approximating this definite integral

I ran into the following integral in my research that I believe has no closed-form solution: $$ I = \int_{s_0}^{s_1} \frac{(\alpha_x s + \beta_x)^{\lambda_x}}{(\alpha_y s + \beta_y)^{\lambda_y}} ds ...
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1answer
36 views

How to evaluate this definite integral which involves sine

The input signal for a given electronic circuit is a function of time $V_{in}(t)$. The output signal is given by $$V_{out}(t) = \int_0^t \sin(t-s)V_{in}(s)ds$$ Find $V_{out}(t)$ if $V_{in}(t) = ...
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1answer
74 views

How to evaluate a definite integral involving the product of two sines?

The input signal for a given electronic circuit is a function of time $V_{in}(t)$. The output signal is given by $$V_{out}(t) = \int_0^t \sin(t-s) V_{in}(s)\,ds$$ Find $V_{out}(t)$ if ...
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1answer
50 views

How to integrate $\sqrt{1+(2/3)x}$?

How would you solve the following (step by step please!): $$\int^6_5\sqrt{1+\frac23x}\ dx$$ I started with $u=1+\frac23x$, $du=\frac23\,dx$, now what?
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3answers
45 views

Improper integral of rational function $k^2/(1+a^2k^2)^2$

I've got the integral $\int^\infty_{-\infty} dk \frac{k^2}{(1+a^2 k^2)^2}$ where $a$ is a real number. I can't seem to find a $u$-substitution or trigonometric substitution that will work. Any ...
0
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2answers
169 views

Volume of Solid Revolution

region bounded by $$y=x$$ and $$y=x^2$$ a) find the volume of the solid of revolution formed by revolving R region about the line $x=2$. b) find the volume of the solid of revolution formed by ...