Questions about the evaluation of specific definite integrals.

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Analysis: Proof checking and help on 2nd part (Integrals)

So I have the question, $f(x)= x$ if $0$ $\leq$ $x$ $\leq$ $1$ and $f(x)= x+2$ if $1<x$ $\leq$ $2$ (the same f(x) I just couldn't figure out how to do the big bracket) Part 1 is asking me to ...
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0answers
3 views

solution to curve limits (concept question)?

I had a question about how the limits work in that 4pi would not give the correct circle distance. I understand that if it has a radius 1 that the distance would be farther but that is only for a ...
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1answer
23 views

Trigonometric Integration: Using the half-angle formula?

I'll preface my question by saying this is my first ever post. I've been lurking around and answering a couple logic questions here and there, but since I have an intractable calculus question I ...
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1answer
30 views

Integral using Beta Function and Gamma Function

Interestingly, I seem to have an integral I have posted before, but I want to take a different approach to it. $\int_{0}^{1} \frac{\ln(1+x)}{1+x^2} \,dx$ The beta function states, $B(x,y) = ...
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1answer
22 views

Volume of an Ellipsoid by double integral

I was finding the volume of the ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ and the answer is supposed to be $\frac{4}{3}\pi abc$, but I am repeatedly getting $\frac{4}{3}\pi ac$. I ...
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1answer
28 views

Evaluating $\int^1_0 \frac{\operatorname{Li}_2(-x)}{x} \log1+(x)\, \mathrm dx$

How would you solve the following? $$\int^1_0 \frac{\operatorname{Li}_2(-x)}{x} \log(1+x)\, \mathrm dx$$
2
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2answers
68 views

Prove that $\int_0^\pi\frac{\cos x \cos 4x}{(2-\cos x)^2}dx=\frac{\pi}{9} (2160 - 1247\sqrt{3})$

Prove that $$\int_0^\pi\frac{\cos x \cos 4x}{(2-\cos x)^2}dx=\frac{\pi}{9} (2160 - 1247\sqrt{3})$$ I tried to use Weierstrass substitution but the term $\cos 4x$ made horrible algebraic-forms since ...
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1answer
46 views

Trigonometric Integration.

Q. $$\int _0^{\frac{\pi }{4}}\:\left(\frac{1}{\left(\cos^4x-\cos^2x\sin^2x+\sin^4x\right)}\right)\:dx$$ My method: =>$$\int _0^{\frac{\pi ...
3
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1answer
46 views

Integral ${\large\int}_0^1\left(-\frac{\operatorname{li} x}x\right)^adx$

Let $\operatorname{li} x$ denote the logarithmic integral $$\operatorname{li} x=\int_0^x\frac{dt}{\ln t}.$$ Consider the following parameterized integral: $$I(a)=\int_0^1\left(-\frac{\operatorname{li} ...
5
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3answers
151 views

Integral of greatest integer function divided by an exponential

If $\lfloor x \rfloor$ denotes the greatest integer not exceeding $x$, then find $\displaystyle\int_{0}^{\infty} \displaystyle \frac{\lfloor x \rfloor}{e^{x}} dx$. The correct answer is supposed to be ...
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0answers
18 views

How to calculate the integral $\int_{\mathbb R^n} e^{i\lambda |x|^\alpha + ix\cdot\xi} dx$?

I want to calculate the $n$-dimensional Fourier transform of the function $e^{i\lambda |x|^\alpha}$, where $\lambda\in\mathbb R$ and $\alpha \in \mathbb R$, that is, the value of the following ...
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1answer
12 views

Find the area of the surface obtained by rotating the curve about the x-axis?

Given this curve: $$y=\frac{x^3}{6}+\frac{1}{2x} 1/2 \le x \le 1 $$ This is what I get for my (dy/dx)^2: $$\frac{x^4+x^{-4}+2}{4}$$ I'm unsure about this. Can anyone confirm that I did it ...
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2answers
107 views

Integral difficulties (attempt included)

I am having difficulties with the following integral. I began working on it and thought I had obtained the answer, but when I went to graph it I received an integral of 1. I obtained the same answer ...
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2answers
63 views

Evaluate $\int_{0}^{\frac {\pi}{3}}x\log(2\sin\frac {x}{2})\,dx$

Prove $$\int_0^{\pi/3}x\log \left(2 \sin\frac {x}{2}\right)\,dx = \frac {2\zeta(3)}{3}-\frac {\pi^2}{9}\log (2\pi)+\frac {2\pi ^2}{3}\log \left|\frac {\Gamma_2 \left(\frac {5}{6}\right)}{\Gamma_2 ...
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4answers
51 views

$\int\frac{2x+1}{x^2+2x+5}dx$ by partial fractions

$$\int\frac{2x+1}{x^2+2x+5}dx$$ I know I'm supposed to make the bottom a perfect square by making it $(x+1)^2 +4$ but I don't know what to do after that. I've tried to make $x+1= \tan x$ because ...
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2answers
26 views

Approximation involving Gamma function: $\frac{\Gamma(j+d)}{\Gamma(j+1)\Gamma(d)}\approx(j-1)^{d-1}$

With $d\leq 1$ and $$ a_j=\frac{\Gamma(j+d)}{\Gamma(j+1)\Gamma(d)}=\frac{d(d+1)\ldots(d+j-1)}{j!},\quad j=0,1,2,\ldots $$ my professor wrote in class that $$ \sum_{j=N}^\infty ...
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2answers
34 views

Find the exact area of the surface obtained by rotating the curve about the x-axis?

So given this curve: $$y=\sqrt{9x-18},\ \ 2 \le x \le 6$$ And using this lovely formula: $$\int2πy\sqrt{1+(\frac{dy}{dx})^2} dx$$ This is what I get for a set up: $$\int_2^62π ...
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2answers
87 views

$\int_{0}^{\infty} \frac {\sqrt x}{x^2-5x+6} \,dx = \frac {\pi}{i \sqrt 3 +i \sqrt 2}$ [on hold]

Prove $$\int_{0}^{\infty} \frac {\sqrt x}{x^2-5x+6} \,dx = \frac {\pi}{i \sqrt 3 +i \sqrt 2}$$ [Nevermind!]: It has been said, it can be solved in at least three ways.I'm looking forward to seeing two ...
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2answers
219 views

Finding the length of a curve?

With the information given: $$x=\frac{y^4}{8}+\frac{1}{4y^2}\,,\ \ 1 \le y \le 2$$ I must find the exact length of the curve. I use this formula to find it: ...
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2answers
145 views

Closed form of $\int_0^1(\ln(1-x)\ln(1+x)\ln(x))^2\,dx$

I remember that some time ago I was asking this question Evaluate $\int_0^1\ln(1-x)\ln x\ln(1+x) \mathrm{dx}$ , and now, while I was making a review, I asked myself if we can get the closed form of ...
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4answers
42 views

Definite integrals involving $\ln x$

Alright, I have been working on this definite integral for the past couple days now and I can't for the life of me obtain the correct answer. I am not too sure where I am going wrong but I think the ...
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3answers
25 views

Finding Length of Arc

This question looked pretty easy at the start, but I did it in a different method as to what the Answerbook has done. And I couldnt get it right. And the answer in the book gave the correct value ...
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2answers
20 views

Stuck on an indefinite integral probably using hyperbolic substitution.

First off, please don't give the answer. I'm really after a starting point. I'm trying to solve the integral $$\int \frac{1}{25e^x+9}~dx$$ I have done a few others where I have an $x$ instead of an ...
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1answer
51 views

Integral of voltage, $\int_{-a}^a \frac{dy}{\sqrt{x^2 + y^2}}$

This is (probably) a very easy integral to solve, but for some reason the answer just isn't coming to me (or at least the one my professor got isn't). He gave us a formula for voltage along the x-axis ...
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0answers
22 views

Evaluating this kind of integral [on hold]

$\int_A^B3ydx+5x^6dy$ along the curve $y=x^2$ and between the point $A$ with coordinates $(0,0)$ and the point $B$ with coordinates $(1,1)$.
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DEFINITE INTEGRAL Caution

Sir/Mam I need to know all different types of precautions that we need to keep in mind while evaluating definite integrals using substitution method. Are there other precautions also to be taken. It ...
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3answers
54 views

Integrating $\displaystyle\int_0^{\pi/2} {\sin^2x \over 1 + \sin x\cos x}dx$

We need to evaluate $\displaystyle \int_0^{\pi/2} {\sin^2x \over 1 + \sin x\cos x}dx$ and some solution to this starts as, $\displaystyle\int_0^{\pi/2} {\sin^2x \over 1 + \sin x\cos x}dx = ...
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0answers
25 views

Evaluating an integral (contour integration probably)

Could you please help me to evaluate the following two integrals? I am looking for the solution of $$\int_{}^{} \frac{A-B\cos(kt)}{\left ( C-D\cos(kt) \right )^{3/2}} dt$$ and $$\int_{}^{} ...
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1answer
37 views

What is the infinite sum of logarithm lnK divided by k(K+1)?

I was trying to calculate the following integral: $\displaystyle\int_1^\infty\frac{dx}{x \lfloor x \rfloor}=? $ which I found to be equivalent to $\displaystyle\sum_{k=2}^\infty\frac{\ln(k)}{k(k+1)} ...
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2answers
62 views

Limit $\lim_{n \rightarrow \infty} \left( \frac{n}{4} \int_0^{1/n}\left(\pi^{\sin^2x}+e^{\sin^2x}\right)^2\cos2x~dx \right)$

Are these statements about the following limits true? $$ \lim_{n \rightarrow \infty} \left( \frac{n}{4} \int_0^{1/n}\left(\pi^{\sin^2x}+e^{\sin^2x}\right)^2\cos2x~dx \right) \stackrel{?}{=} 1, $$ $$ ...
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1answer
59 views

Can I use the residue calculus here?

I know from an alternative to the residue calculus that $$I=\int_{-\infty}^{\infty}\frac{e^x}{e^{e^x}+1}dx = \log 2.$$ However, I see no reason I cannot apply the residue calculus here. My attempt: ...
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2answers
27 views

Simplifying a sigma expression with square roots in the denominator?

Let $$A = \frac{1}{\sqrt{n}\sqrt{n + 1}} + \frac{1}{\sqrt{n}\sqrt{n + 2}}+...+\frac{1}{\sqrt{n}\sqrt{n + n}}$$ where $n$ is any real number such that, $$A = \sum_{i = ...
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0answers
39 views

Closed-form solution for the integral of the ratio of two sums?

Is there a closed-form solution for this integral? $$\int_0^t\frac{\sum_{k=1}^Nr_{1k}\exp(ixr_{2k})}{\sum_{k=1}^N\exp(ixr_{2k})}dx$$ for given $r_{11},...,r_{1N},r_{21},...,r_{2N} \in \mathbb{R}$
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1answer
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Tripe integral, with the integration of $\sin(x+y^2)$ [closed]

how to calculate the integral below: $$I=\iiint_{\Omega} |\sin(x+y^2)|dxdydz$$ where $\Omega$ is the zone whose edge is $x=0$,$y=0$,$x=\pi$,$y=\pi/2$ and $z=y$. Thanks very much.
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3answers
69 views

Is $f(x)=x$ the solution of an integral equation? [closed]

Suppose that $f:[0, \infty)\longrightarrow \mathbb{R}$ is continuous and $f(x) \neq 0 $ for all $x>0$. If $$ \big(\,f(x)\big)^2=2 \int_0^x f(t)\,dt, $$ for all $x>0$, is it then true that ...
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2answers
25 views

Multiple answer for integration of a function?

Q. $\int \left(\frac{sin2x}{sin^4x+cos^4x}\right)\:dx$ My method: $$\int \:\left(\frac{sin2x}{sin^4x+cos^4x}\right)\:dx=\int ...
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1answer
17 views

Finding volume of solid using two methods

The question is: The region bounded by $y=\frac{1}{x}, y=0, x=1, x=2$ is rotated about the $y$-axis, thus creating a solid. Compute the volume using the Shell and Slicing method. This is what I have ...
2
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2answers
39 views

Integral of the product of squared exponential and two erf functions

I'm trying to solve the following integral $$ \int_{-\infty}^{\infty} e^{-(\alpha t + \beta)^2}\operatorname{erf}(at + b)\operatorname{erf}(ct + d)\text{d}t $$ I've tried with differentiation under ...
3
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1answer
53 views

Stuck on tough integral

I am trying to solve what looked like a simple integral but I got a bit stuck. The integral is : \begin{equation} \int_0^x \frac{ab(1-e^{-ct})}{d-\frac{b(1-e^{-ct})}{c}}dt \end{equation} I tried ...
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3answers
74 views

How to evaluate the following? $\sum\limits_{n=1}^{\infty} \int_1^{n!} x^{-n}{\rm d}x$.

How can I evaluate this sum? $$\sum_{n=1}^{\infty} \int_1^{n!} x^{-n} \mathrm{d} x$$ Is there a closed form or a transform that would make it possible not to "CAS" it? Also, it seems to converge, but ...
16
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3answers
185 views

Finding integral of a function

I have stumbled upon an exercise that reads thus: $$\int\limits_{-\infty}^0\frac{x^x}{x^3-1}\mathrm{d}x=\frac{2\sqrt3}{9}\pi,$$ and I am guessing it is asking to prove the above equality. ...
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1answer
30 views

Setting up integral for volume of the solid

How would I set up an integral for the volume of the solid bounded between these two curves: $$y=x$$ $$y=\frac{2x}{1+x^3}$$ Rotated about x=-1. And these two curves: $$y^2 - x^2 = 1$$ $$y=2$$ ...
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0answers
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+50

Closed form of integrals containing double exponentials

Are there closed forms for the following integrals? $$\begin{align} I_1(w) & = \int_{-\infty}^{\infty} \frac{\exp(-we^y)}{y^2+\pi^2} dy, \\ I_2(w) & = \int_{-\infty}^{\infty} ...
2
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1answer
40 views

$\int_0^{2 \pi} \cos(x)e^{i (a \cos(x) + b \cos^2(x)} dx$ and $\int_0^{2 \pi} \cos^2(x)e^{i (a \cos(x) + b \cos^2(x)} dx$

I am currently dealing with the two integrals in the title and I want to find out, when their real part of their imaginary part vanishes ( so for which constellation of $(a,b) \in \mathbb{R}^2 ...
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2answers
101 views

How to find this integral $I=\int_{-\pi}^{\pi}\frac{x\cdot \sin(x) \cot^{-1}{(2014^x)}}{1+\cos^4(x)}dx$

Question: Find this integral $$I=\int_{-\pi}^{\pi}\frac{x\cdot \sin(x) \cot^{-1}{(2014^x)}}{1+\cos^4(x)}dx$$ let $x\to -x$,so $$I=\int_{-\pi}^{\pi}\dfrac{x\sin(x) ...
68
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8answers
3k views

Really advanced techniques of integration (definite or indefinite)

Okay, so everyone knows the usual methods of solving integrals, namely u-substitution, integration by parts, partial fractions, trig substitutions, and reduction formulas. But what else is there? ...
4
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1answer
47 views

Integral $a\int_{-\infty }^{\infty } \frac{e^{\frac{x^2}{a^2+x^2}}}{a^2+x^2} \,\operatorname dx$

I'm looking to calculate $a\int_{-\infty }^{\infty } \frac{e^{\frac{x^2}{a^2+x^2}}}{a^2+x^2} \,\operatorname dx$. Mathematica10 can't integrate this, but numerical integration gives an answer of ...
3
votes
0answers
97 views

Calculation of this integral

Evaluating $$\int_0^1\frac{\log^2(x)\log(1+x^2)}{1-x^2}dx$$ I found $- \dfrac{\pi^4}{32}+2G^2+\dfrac74 ζ(3)\log2 $ where G is the Catalan's constant.
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1answer
28 views

Calculating area relative to the y-axis

I was asked to calculate the area of the region bounded by the following graph: $$ y = x^2+4x ; y=0$$ I substituted $y$ in order to get $x = 0$ 0r $x=4$. Now I would like a little bit of help to get ...
6
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2answers
82 views

Calculating value of $\pi$ independently using integrals.

Recently I noticed this integral: $$\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}7-\pi\approx0$$ Which is a very interesting result which gives us the value of ...