Questions about the evaluation of specific definite integrals.

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1answer
9 views

error order evaluation in taylor expansion of a definite integral

I have a function $g(x)=f(x)e^{-x}$ and i want to consider the following integral: $\int_{0}^{\infty}g(x)dx$. Since $f(x)$ is a complicated, but monotonic decreasing, function in the interval ...
0
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0answers
27 views

integration involving imaginary terms

How do we integrate forms of following type with imaginary terms involved? Can we get a closed form of it as result? $\int -\frac{1}{4} (e^{-ix}-e^{ix})^2 e^{\frac{4i (e^{-ix}-e^{ix}) }{ ...
1
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0answers
31 views

Integration using exponent

What could be the techniques we need to use to solve this integration $\displaystyle \int tan^2 \theta \frac{\sin^2\left( sec \theta \hspace{.1cm}tan\hspace{.1cm} \theta \right) }{ sec^2 \theta } ...
0
votes
0answers
12 views

Calculus Single Variable: Find max and min of hard to graph function

Consider the function F defined by F(x)= integral from 0 to x of $t|sint(t)|dt$. Find the absolute maximum value and absolute minimum value of y=f(x). I know there's one at x= zero but the ones ...
7
votes
0answers
50 views

Integral $\int_0^1\frac{\log(x)\log^2(1-x)\log^2(1+x)}{x}\mathrm dx$

I decided to follow a recent trend and ask a question about logarithmic integrals :) Is there a closed form for this integral? $$\int_0^1\frac{\log(x)\log^2(1-x)\log^2(1+x)}{x}\mathrm dx$$
1
vote
1answer
21 views

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} ...
1
vote
1answer
12 views

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 ...
7
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1answer
41 views

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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2answers
117 views

Evaluating $\int_{0}^{\pi/3}\ln^2 \left ( \sin x \right )\,dx$

Good evening! I want to compute the integral $\displaystyle \int_{0}^{\pi/3}\ln^2 \left ( \sin x \right )\,dx$. However I find it extremely difficult. What I've tried is rewritting it as: ...
1
vote
0answers
46 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 ...
0
votes
1answer
35 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: $$ 2\cos7x=2-2\cos7x $$ $$4\cos7x=2$$ $$\cos7x=1/2$$
9
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0answers
74 views

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? ...
2
votes
1answer
45 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 ...
3
votes
4answers
223 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 ...
0
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0answers
30 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. ...
0
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1answer
43 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 ...
-1
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2answers
32 views

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
1
vote
1answer
36 views

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 ...
0
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3answers
41 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 ...
0
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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 ...
1
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4answers
83 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 ...
0
votes
0answers
55 views

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 ...
2
votes
1answer
31 views

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 ...
1
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2answers
48 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 ...
17
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2answers
266 views
+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}] ...
2
votes
0answers
33 views

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} ...
4
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0answers
197 views
+300

Integration of product of functions(Special form)

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 ...
6
votes
6answers
192 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 ...
5
votes
2answers
54 views

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 ...
0
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4answers
151 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
0
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0answers
40 views

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: ...
15
votes
3answers
248 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 ...
17
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3answers
284 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 ...
2
votes
4answers
111 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 $\dfrac{\pi^4}{15}$
0
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3answers
43 views

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 ...
0
votes
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 ...
6
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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 ...
2
votes
0answers
42 views

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 ...
4
votes
3answers
86 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 ...
2
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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 ???!
0
votes
1answer
35 views

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) &= ...
3
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0answers
33 views

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 ...
7
votes
0answers
164 views

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 ...
3
votes
1answer
32 views

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) ...
1
vote
2answers
72 views

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 ...
1
vote
3answers
40 views

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. $$ ...
0
votes
2answers
33 views

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 ...
-5
votes
0answers
40 views

How do I do this [closed]

I want the steps (and possibly proof for any answer you provide) to solve this problem!
0
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1answer
40 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 ...