Tagged Questions

Questions about the evaluation of specific definite integrals.

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2
votes
0answers
26 views

Finding the closed form of $\int_0^1 \frac{(1-x+x\log(x))\operatorname{Li_3(x)}}{(x-1) x \log(x)} \ dx$

Here I have a question I just received, and still trying to find a proper starting point $$\int_0^1 \frac{(1-x+x\log(x))\operatorname{Li_3(x)}}{(x-1) x \log(x)} \ dx$$ What starting point would you ...
8
votes
0answers
48 views

Evaluation of a tough double integral

This is an integral coming from personal research, and very important to me, but it does not seem an easy job to do. If a solution is not possible then I'd be glad with a closed form only. ...
1
vote
0answers
7 views

How to write upper and lower generic limits of a definite integral

I have an integral whose upper and lower limits depends on the probabilty distribution chosen for the variable, $x$, I am integrating on. For example, if I consider that $x$ is normally distributed ...
0
votes
1answer
22 views

Am I correct with this change of variable?

I have been solving a problem from a paper I read related to poisson point processes and for some reason I am not reaching the same result the paper has. The problem is re-expressing an expression by ...
-2
votes
1answer
24 views

Volume by Washers and Shells [on hold]

bounded by $y=x^2$, $y=4-x^2$, rotated about the x-axis bounded by $x+y=1$, $x-y=-1$, and the line $x=2$, rotated about the y-axis. Bounded by $x+y=3$, $y+x^2=3$, rotated about the line $x=2$. ...
0
votes
1answer
20 views

Find the volume of the solid generated by revolving the region bounded by $y=x$ and $y=x^2$ about the line $y=x$

Find the volume of the solid generated by revolving the region bounded by $y=x$ and $y=x^2$ about the line $y=x$ I am confused, how do we approach such problems, where the rotation lines are not ...
0
votes
1answer
20 views

Volume of a solid sphere hole

A round hole of radius $\sqrt{3}$ is bored through the centre of a solid sphere of radius 2. Find the volume of the material removed . Looking for a clever way to solve this problem
0
votes
0answers
83 views

Nasty integration?

So I am trying to solve the following integral and apparently its not integrable or I might be wrong. Not even computer software can integrate. Can anyone tell me if this is integrable or not? The ...
1
vote
4answers
49 views

Derivative of $\int_{x}^{0} \frac{\cos xt}{t} dt$

I am working on the following problem: Find the derivative of $f(x)=\displaystyle \int_{x}^{0}\displaystyle\frac{\cos xt}{t}dt$. The answer I am supposed to get is $\displaystyle ...
1
vote
1answer
20 views

troublesome area problem

Calculate the area of the region of the graph bounded by: $$\begin{eqnarray} y &=& x \\ y &=& x^2 + 1 \\ y &=& 2 \\ x &=& 0 \end{eqnarray}$$ My final result is ...
0
votes
0answers
20 views

Finding an integral for a given Riemann Sum

Take the Riemann sum: $= \displaystyle \lim_{m\to\infty} \frac{1}{m}\sum_{x=1}^{m} me^{-x}$ How can someone convert that into an integral? We know $\Delta(x) = \frac{1}{m}$. So $me^{-x}$, is the ...
2
votes
0answers
67 views

Is there a function whose definite integrals are all 0?

Is there a continuous function $f: [0,1] \rightarrow \mathbb{R}$ such that $f(x) \neq 0$ for some $x \in [0,1]$ and, if we define $F_n(x) = \int_{0} ^ {x} F_{n-1}(t) dt $ (where $F_0(x)=f(x)$), then ...
2
votes
1answer
45 views

Why am I obtaining an imaginary part for my integration

I try to solve an integration as follows, $$\int \frac{sy^{-1}}{(1+sy^{-1})} \text{exp}(-\sqrt{y})dy$$ as you can see its pretty complicated, and I get an answer like the following using Wolfram ...
10
votes
1answer
108 views

Evaluating by real methods $\int_0^{\pi/2} \frac{x^5}{2-\cos^2(x)}\ dx$

I'm sure you guys can briefly get the result by some methods of complex analysis, but now I'm only interested in real analysis methods of proving the result. What would you propose for that? ...
0
votes
2answers
35 views

U-Substitution. Why do you multiply the integrand by -1 in this case?

$$\int_0^{\pi/2} \! \frac{\sin x\cos x}{(4-\sin^2 x)^2} \:\text{d}x$$ set $u = 4-\sin^2 x$, therefore $du = -2 \sin x \cos x \text{d}x $ $$-\frac{1}{2} \int u^{-1/2} \text{d}u $$ Change the range ...
1
vote
1answer
56 views

Tough definite integration

For a curve given by: $x=e^{-t}\cos{2t}$, $y=\sin t$ R is the region bounding this curve, the x axis and the y axis (y-intercept is point a and x-intercept is point b). Find the exact coordinates ...
3
votes
0answers
56 views

Solving double integrals numerically?

I have written this in way to make it as much as possible non-confusing. I will start describing my problem and I will walk you through my question, I have a double integration which I am trying to ...
4
votes
0answers
87 views

How to compute or simplify this nasty integration?

Any hints on solving an integration of the following form, $$\int_{x}^{+\infty}\left(1-\frac{1}{1+sy^{-1}}\right) \left(\text{exp}(-\sqrt{y})+ y^{-\frac{1}{2}}(1-\text{exp}(-\sqrt[4]y)\right)dy $$ ...
9
votes
1answer
85 views

A limit evaluating to $2 K$ (Catalan's constant)

Experimentally I discovered the limit below that says that $$\lim_{n\to\infty} \int_0^{\pi/2} \frac{1}{\displaystyle ...
0
votes
1answer
25 views

How to find a bound for these (simple) integrals

With help of $\int_{0}^{\infty} e^{-x^2} dx = \frac{\sqrt{\pi}}{2}$ and $\int_{0}^{\infty} e^{-x} dx =1$, I would like to know how to derive the following bounds: $$\int_{0}^m 4e^{-\frac{t^2}{8m}}dt ...
20
votes
2answers
187 views

Prove the integral evaluates to $\frac{K}{\pi}$

Yesterday I received the following integral that might require some tedious steps to do $$\int_0^{\infty}{\small\left[ \frac{x}{\log^2\left(e^{\large x^2}-1\right)}- \frac{x}{\sqrt{e^{\large ...
0
votes
2answers
26 views

Integrated series identity with Legendre Polynomials

The Legendre Polynomials can be defined in many different ways and have several properties. Many of these can be found in books or in the net, but I couldn't find this one anywhere: Prove that: ...
0
votes
0answers
25 views

Help in evaluating the integral at the given limits

Hi guys I am hoping to get some help here. The indefinite integral below gives the following result. $$\int \left(\frac{0.0016 \left(1-\exp \left(-0.0112 v^{0.25}\right)\right)}{v^{0.5}}+\frac{0.0036 ...
0
votes
1answer
26 views

laplace method on this integral

How to get the leading asymptotic expansion for this integral $\int_{0}^{\pi/2}\sqrt{\sin(t)}\exp(-x\sin^4(t))dt $ in the limit $x\rightarrow\infty$ ? Because the maximum of the exponent is at $t=0$ ...
7
votes
3answers
83 views

Finding the integral $\int_{0}^{\infty }\frac{1}{(4x-3)(4x-1)}\,dx$.

Which method that will be effective for solving this integral?
4
votes
0answers
95 views
+100

$\int_{-\infty}^\infty \exp (a t^3 + b t^2 + c t) \mathrm{d}t,\;\;(a,c)\in\mathbb{I}, \; \; \Re(b)\le0$

$$\int_{-\infty}^\infty \exp (a t^3 + b t^2 + c t) \mathrm{d}t,\;\;(a,c)\in\mathbb{I}, \; \; \Re(b)\le0$$ i.e. an oscillation with frequency $3\Im(a)t^2 + 2\Im(b)t + \Im(c)$ and phase $0$, multiplied ...
3
votes
2answers
68 views

Evaluation of $\int_0^\pi \! \ln\left(1-2\alpha\cos x+\alpha^2\right) \mathrm{d}x$

I have got a trouble with integral $$\int_0^\pi \! \ln\left(1-2\alpha\cos x+\alpha^2\right) \, \mathrm{d}x,\quad |\alpha|<1.$$ My teacher said there are two ways of solving such ones, if there is ...
2
votes
1answer
28 views

Trigonometric Integrals times exponential

Let's say I want to evaluate the integral: $$\int_0^{\pi/2} e^{ax}\cos^{a}(x) \,dx$$ where $1\leq a \leq 5$ . One standard way to go around would be by applying parts. That would result of course in ...
5
votes
5answers
184 views

Intuitive explanation of integral identity

I have been able to prove the identity $$\int_{0}^{1} \frac{f(x)}{f(x)+f(1-x)} \, dx = \frac{1}{2}$$ for any continous $f:[0,1]\to[0,\infty)$ for which the integrand is defined, with calculus, but I ...
1
vote
3answers
73 views

Solve $ \int_0^{\sqrt{\pi / 2}}\left(\int_x^{\sqrt{\pi / 2} }\sin(y^2) dy \right)dx$

I'm trying to solve this: $$ \int_0^{\large\sqrt{\frac{\pi}{2}}}\left(\int_x^{\large\sqrt{\frac{\pi}{2}}}\, \sin y^2\, dy \right)dx $$ But I'm having trouble with finding an primitive to $\sin(y^2)$. ...
0
votes
0answers
22 views

Does it really matter that we are using the Taylor polynomial and remainder?

Assuming that the quadrature rule $I_n$ integrates all polynomials of degree less than or equal to N exactly: $I_n(p)$=$I(p)$ for all p $\epsilon$ $P_N$. Using this it could be proved that for any ...
0
votes
2answers
27 views

integrate sine at denominator

This integral: $\int_{\pi/2}^0\frac{d\theta}{1-\gamma\sin 2\theta}$. I tried $e^{i\theta}=\cos\theta+i\sin\theta$ and $\sin^2\theta+\cos^2\theta=1$, but didn't succeed. Is there any one can help me? ...
1
vote
1answer
49 views

Help in computing this integration?

Any thoughts or hints on solving the following integral $$ \int_{y}^{+\infty} \frac{ e^{-\sqrt{v}} }{1+ s v^{-1} } dv $$ and where $$s= \frac{2}{x^{-1}+y^{-1}}$$ The result should be a function ...
9
votes
3answers
88 views

How to prove $\int_0^{2\pi} \ln(1+a^2+2a\cos x)\, dx=0$.

How can I prove $\int_0^{2\pi} \ln(1+a^2+2a\cos x)\, dx=0$, where $a<1$? Thanks.
0
votes
2answers
31 views

Help with the integral $\int_{-b/2}^{\frac{\pi-10b}{20}}\frac{\mathrm{d}x}{x^2+bx+c}$

Let $b$ and $c$ satisfy the equation $4c-b^2 = \frac{\pi^2}{100}$, then how should I solve the integral: $I =\int_{-b/2}^{\frac{\pi-10b}{20}}\frac{\mathrm{d}x}{x^2+bx+c}$ All I know is that the ...
10
votes
2answers
97 views

Evaluating $\int_0^{\infty}\left(\frac{\ln(1+x^2)}{1+x^2}\right)^3 dx$

I’m looking for a closed form of this integral $$I_3=\int_0^{\infty}\left(\frac{\ln(1+x^2)}{1+x^2}\right)^3 dx$$ I’ve managed to evaluate $$I_1=\int_0^{\infty} \frac{\ln(1+x^2)}{1+x^2}dx=\pi \ln 2$$ ...
0
votes
0answers
22 views

Finding the inverse of an integral

I'm looking for a computational approach here, since I don't think there is a closed-form solution. I have the following: $$ s(x) = \rho + \int_{\rho}^{x} \sqrt{ 1 + (\alpha \cos t - k)^2 } \, dt $$ ...
4
votes
1answer
72 views

integral of a function

I wanted to find the integral of the function $f(x)$ from zero to one: $$f(x)=\begin{cases}2x\sin(1/x)-\cos(1/x) & : x\in(0,1]\\ 0 & :x=0\end{cases}$$ but I think whether its integral is not ...
4
votes
1answer
35 views

Three integral involving polylogarithm function

$\newcommand{\Li}{\operatorname{Li}}$Evaluate the following integrals $$\int\limits_0^1 \frac{\Li_2^3(x)}{x}dx, \quad \int\limits_0^1 \frac{\Li_2^2(x)\Li_3 (x)}{x} dx, \quad \int\limits_0^1 ...
2
votes
1answer
33 views

Integration of complex functions

Show that $\displaystyle \int_{-\infty}^\infty x^{2n}e^{-x^2}dx=(2n)!\frac{\sqrt\pi}{4^n n!}$ by differentiating the equation $\displaystyle \int_{-\infty}^\infty e^{-tx^2} dx=\sqrt{\frac{\pi}{t}}$. ...
0
votes
1answer
46 views

rational exponent of negative base

I have the definite integral $$\int_{1}^{\,9} {\frac{6}{\sqrt[3]{x-9}}}\, \mathrm dx$$ When I try to evaluate it I get the indefinite integral equals $9(x-9)^{2/3}$ and evaluating at the limits gives ...
2
votes
1answer
148 views

How to evaluate $ \int_{0}^{1} x^{x^{x^{x^…}}} dx $

Inspired by the fact that $\int_0^1 \frac{1}{x^x}=\sum_{k=1}^\infty \frac{1}{k^k}$ I asked myself wether it is possible to evaluate the following integral: $$ \int_{0}^{1} x^{x^{x^{x^…}}} dx $$ In a ...
9
votes
2answers
177 views

Closed form of $\int_0^{\infty} \frac{\log(x)}{\cosh(x) \sec(x)- \tan(x)} \ dx$

What real analysis tools would you recommend me for getting the closed form of the integral below? $$\int_0^{\infty} \frac{\log(x)}{\cosh(x) \sec(x)- \tan(x)} \ dx$$
7
votes
2answers
147 views

Using differentiation under integral sign to calculate a definite integral

I want to calculate the integral $$\int^{\pi/2}_0\frac{\log(1+\sin\phi)}{\sin\phi}d\phi$$ using differentiation with respect to parameter in the integral ...
-4
votes
0answers
57 views

Show $\int_0^{\pi/2}\ln\biggl(\frac{\ln^2(\sin x)}{\pi^2+\ln^2(\sin x)}\biggr)\frac{\ln(\cos x)}{\tan(x)}dx=\frac{\pi^2}{4}$ [duplicate]

Yesterday I found this integral on Quora: How does one prove the following integral? $$ \int_0^{\pi/2}\ln\biggl(\frac{\ln^2(\sin x)}{\pi^2+\ln^2(\sin x)}\biggr)\frac{\ln(\cos ...
4
votes
3answers
122 views

Prove $\int_0^{2\pi}\frac{3a\sin^2\theta}{(1-a\cos \theta)^4}$ or $\int_0^{2\pi}\frac{\cos\theta}{(1-a\cos\theta)^3}=\frac{3a\pi}{(1-a^2)^{5/2}}$

While doing some mathematical modelling of planetary orbits I have come up with two definite integrals $D_1$ and $D_2$ which appear to produce the same result R when tested with various values of $a$ ...
3
votes
0answers
99 views

To determine a definite integral [duplicate]

I have been trying to solve the following integral $$\int_{0}^{\frac {\pi}{2}} \ln\left (\frac {\ln^2 (\sin x)}{\pi^2+\ln^2 (\sin x)}\right) \frac {\ln \cos x}{\tan x} dx$$ I tried substituting for ...
0
votes
1answer
27 views

Change of limits in definite integral - non constant limit

I have the following definite integral: $\displaystyle \int^{g(a)}_{a} f(x) \, dx$ and I am asked to perform a shift of the variable x, so that it transforms in $x + T$ (T is just some constant). ...
14
votes
1answer
141 views
0
votes
0answers
24 views

volumes and integrals help [closed]

Using disks or washers, find the volume of the solid obtained by rotating the region bounded by the curves $y=x^2$ and $y=x^4$ and $y=16$ and $x=0$ about the $y$-axis.