1
vote
2answers
21 views

Recursive formula for definite integral

The integral is: $$I_n = \int_0^{\pi/4} \tan^{2n}x\,dx$$ I'm supposed to find the recursive formula.
4
votes
0answers
57 views

Delicate Integral $I=\int_0^\infty \frac{\log^2 x \cos ax}{x^n-1}dx$

Hi I am trying to calculate $$ I:=\int\limits_0^\infty \frac{\log^2 x \cos (ax)}{x^n-1}dx,\quad \Re(n)>1, \, a\in \mathbb{R}. $$ Note if we set a=0 we get a similar integral given by $$ ...
4
votes
0answers
48 views

Fancy Integral $I=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx$

Hi I am trying to integrate and obtain a closed form result for $$ I:=\int_0^1 \frac{\log x \log (1+x) \log(1-x) \log(1+x^2)\log(1-x^2)}{x^{3/2}}dx. $$ Here is what I tried (but I do not think this is ...
10
votes
1answer
235 views

Cool Integral = $\pi/2$ !!

I am trying to calculate the integral $$ I_n=\int \limits_0^\infty \prod_{k=1}^n \frac{\sin \frac{x}{2k-1}}{\frac{x}{2k-1}}dx. $$ (I have literature on this, if people want). Note, we can write the ...
4
votes
0answers
96 views

Beautiful Closed form $\int_0^1 dx \frac{\ln x \ln^2(1-x)\ln(1+x)}{x}$

Hi I am trying to calculate $$ I:=\int_0^1 dx \frac{\ln x \ln^2(1-x)\ln(1+x)}{x}$$ Note, the closed form is beautiful and is given by $$ I=−\frac{3}{8}\zeta_2\zeta_3 -\frac{2}{3}\zeta_2\ln^3 2 ...
3
votes
0answers
35 views

Fundamental Theorem of Calculus and inverse..

If $F(x)$ is defined as $$F(x)= \int_{a}^{x} f(t) dt$$ calculate $(F^{-1})'(y)$ in terms of $f$. I have been working on this for a while now, does the aanswer to this incorporate the Inverse ...
0
votes
1answer
31 views

a question about integral? I have no idea about that!

If f(x) and g(x) are integrable in [a,b], can we say that f(x)g(x) is still integrable in [a,b]? I am referring to Riemann integration!
2
votes
0answers
45 views

Log Cosine Integral $\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3)$

$$ I=\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3). $$ Note $\zeta(3)$ is given by $$ \zeta(3)=\sum_{n=1}^\infty \frac{1}{n^3}. $$ I have a ...
10
votes
0answers
289 views
+400

$I=-\frac{4}{3}\log^3 2-\frac{\pi^2}{3}\log 2+\frac{5}{2}\zeta(3)$

Hi I am trying to evaluate $$ I:=\int \limits_{0}^{1} \left[ \frac{1}{x(x-1)} \bigg(2\mathrm{Li}_2\bigg(\frac{1-\sqrt{1-x}}{2}\bigg)-\log\bigg(\frac{1+\sqrt{1-x}}{2}\bigg)^2 \bigg) ...
7
votes
3answers
111 views

Showing that $\int_{0}^{\infty} \frac{dx}{1 + x^2} = 2 \int_0^1 \frac{dx}{1 + x^2}$

I was reading an article in which it was stated that, with a change of variable, one could show that: $$\int_{0}^{\infty} \frac{dx}{1 + x^2} = 2 \int_0^1 \frac{dx}{1 + x^2}$$ I tried with $t = 1 + ...
2
votes
1answer
53 views

Integral $\int_0^{\pi/2} \log^n (\sin t)\log^p (\cos t) dt$

I am looking for a closed form expression for the logarithmic trigonometric integral $$ I_{n,p}=\int_0^{\pi/2} \log^n (\sin t)\log^p (\cos t) dt \quad (n\geq 0, p\geq 0). $$ Closed form expression ...
10
votes
2answers
163 views

Calculate $\int_0^1 \frac{\ln(1-x+x^2)}{x-x^2}dx$

I am trying to calculate: $$\int_0^1 \frac{\ln(1-x+x^2)}{x-x^2}dx$$ I am not looking for an answer but simply a nudge in the right direction. A stradegy, just something that would get me started. ...
1
vote
1answer
49 views

integral $I=\int_{-\infty}^\infty e^{-\alpha x^{2k}}dx$

$$ I=\int_{-\infty}^\infty e^{-\alpha x^{2k}} dx $$ The last problem was ill posed, and is answered in the post! You can disregard this post!
11
votes
1answer
244 views
+400

$I=\frac{1}{\pi}\int_0^{\pi/3}\log\big( \mu(\theta)+\sqrt{\mu^2(\theta)-1} \big)\ d\theta, \quad \mu(\theta)=\frac{1+2\cos\theta}{2}.$

Hi I am trying to calculate this integral: $$ I=\frac{1}{\pi}\int_0^{\pi/3}\log\left( \frac{1+2\cos\theta}{2}+\sqrt{\bigg( \frac{1+2\cos\theta}{2} \bigg)^2-1} \right)\ d\theta. $$ The ...
2
votes
1answer
34 views

A bounded integral

I want to show that there exists $K\in\mathbb{R}^+$ such that $$\left|\int_{1}^x \sin(t+t^7)dt \right|<K$$ for all $x\ge 1$. Intuitively, I'm quite sure it is true, but I can't find a formal proof. ...
0
votes
0answers
57 views

LogSine Integrals $\int_0^{\pi/3}\theta \ln^2\big(2\sin\frac{\theta}{2}\big)d\theta$.

Hi this will soon end my posts on Log Sine integrals, and we can progress into other classes of integrals. The log sine integral I am trying to calculate is given by $$ ...
1
vote
0answers
45 views

$\frac{5\pi^3}{154}=\int_{\pi/6}^{\pi/2}\bigg[\Re\big(\text{Li}_2(4\sin^2\theta)\big) +\text{Li}_2\bigg(\frac{1}{4\sin^2\theta}\bigg) \bigg]d\theta$

I am trying to prove $$ \int_{\pi/6}^{\pi/2}\bigg[\Re\big(\text{Li}_2(4\sin^2\theta)\big) +\text{Li}_2\bigg(\frac{1}{4\sin^2\theta}\bigg) \bigg]d\theta=\frac{5\pi^3}{54}. $$ Clearly, this closed form ...
1
vote
0answers
45 views

LogSine Integral $\int_0^{\pi/3}\ln^n\big(2\sin\frac{\theta}{2}\big)d\theta$

I am trying to integrate the Log Sine Integral: $$ Ls_{n+1}=-\int_0^{\pi/3}\bigg[\ln\big(2\sin\frac{\theta}{2}\big)\bigg]^nd\theta $$ where n is a non-negative integer. This problem is strongly ...
0
votes
0answers
33 views

LogSine Moments $\int_0^\sigma \theta^k \ln^{n-1-k}\big| 2\sin\frac{\theta}{2}\big|d\theta$

This integral is known as the moments for the generalized log-sine integrals. The notation I am using is similar to Lewin and what he used in the 1950's-1980's. $$ ...
1
vote
0answers
32 views

LogSine Generating Fn $ \int_0^\pi \big(2\sin\frac{\theta}{2}\big)^x e^{\theta y} d\theta$

This is related to generating functions for Ls (Log Sine Integrals.) I am trying to calculate $$ \int_{0}^{\pi}\left[2\sin\left(\theta \over 2\right)\right]^{x} {\rm e}^{\theta y}\,{\rm d}\theta. $$ ...
2
votes
1answer
125 views

LogSine Integral $I=-\int_0^{\pi/3} \ln^2\big(2\cos \frac{\theta}{2}\big) d\theta$

These are known as LogSine integrals at $2\pi/3$, so I will call the integral Ls as this is common in the literature. I am trying to prove $$ Ls=-\int_0^{\pi/3} \ln^2\big(2\cos \frac{\theta}{2}\big) ...
2
votes
2answers
111 views

Integrate $I=\int_0^{\pi/2}x^2\ln(\sinh x)\ln(\cosh x)dx$

Hi I am trying to evaluate the integral $$I=\int_0^{\pi/2}x^2\ln(\sinh x)\ln(\cosh x)dx.$$ Note we can write the integrand as $$ x^2 \ln\big(\frac{e^x-e^{-x}}{2}\big) ...
2
votes
1answer
121 views

Integrate $\int_0^\pi \theta^2 \ln^2\big(2\cosh\frac{\theta}{2}\big)d \theta$

Hello I am trying to integrate $$ I=\int_0^\pi \theta^2 \ln^2\big(2\cosh\frac{\theta}{2}\big)d \theta $$ which is similar to Integral...$\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d ...
2
votes
0answers
58 views

Integral $\int_0^\infty \frac{x^n}{(x^2+\alpha^2)^2(e^x-1)^2}dx$

Hey I am trying to integrate $$ \int_0^\infty \frac{x^n}{(x^2+\alpha^2)^2(e^x-1)^2}dx,\quad \alpha,n \geq 1. $$ Thanks. This integral is old and has a 4th order pole. I am also looking for literature ...
1
vote
0answers
31 views

Integral$\int_{-\infty}^\infty x^{2n} e^{-\beta (x^2+\cos x+\alpha x)}dx$

Hi I am trying to integrate $$ \int_{-\infty}^\infty\int_{-\infty}^\infty (xy)^{2n}\exp\left({-\beta(x^2+y^2+\cos x+\alpha x+iy)}\right)dxdy \quad \alpha,\beta,n >0. $$ These integrals can be ...
1
vote
1answer
38 views

Integral $\int_{-\infty}^\infty dx e^{-nx^2/2}(z-ix)^n$

$$ I\equiv\mathcal{F}_n(z)=\int_{-\infty}^\infty dx e^{-nx^2/2}(z-ix)^n. $$ Evaluate I for $n \to \infty$ and z real. We can consider $z\geq 0$ due to the symmetry of $\mathcal{F}$ given by $$ ...
1
vote
0answers
48 views

Integrate $ \int_0^\infty \frac{x^n\ln x}{(x^2+\alpha^2)^2(e^x-1)}dx$

Hey I am trying to integrate $$ \int_0^\infty \frac{x^n\ln x}{(x^2+\alpha^2)^2(e^x-1)}dx,\quad \alpha,n \in \mathbb{R}^{0+}. $$ This integral is old and has a triple pole. I am also looking for ...
3
votes
1answer
156 views

Integrate $ \int_0^{\pi/2} \frac{x^{2p}}{1+\cos^2x}dx $

Hi I am trying to come up with a closed form expression for $$ \int_0^{\pi/2} \frac{x^{2p}}{1+\cos^2x}dx,\quad p\geq 0. $$ I am interested in this general case in terms of p. For small p, we can ...
1
vote
0answers
52 views

Integrate $ \int_0^{\phi_0} \arctan \sqrt{\frac{\cos \phi+1}{\alpha \cos \phi +\beta}}d\phi$

EDIT/UPDATE: I DO NOT NEED A SOLUTION. SEE SOS440 COMMENT FOR A FULL DETAILED SOLUTION. Hi I am trying to integrate $$ \int_0^{\phi_0} \arctan \sqrt{\frac{\cos \phi+1}{\alpha \cos \phi +\beta}}d\phi, ...
0
votes
0answers
68 views

Integrate $\int_0^\infty \frac{Li_n(-\sigma x)Li_m(-\omega x^2)}{x^3}dx$

I am trying to solve this integral $$ \int_{0}^{\infty} {{\rm Li}_{n}\left(-\sigma x\right){\rm Li}_m\left(-\omega x^{2}\right) \over x^{3}}\,{\rm d}x $$ which is from some high school IMO training ...
3
votes
3answers
62 views

Integrating a function with an infinite number of discontinuities

I would appreciate some help with the following exercise: Let $$f(x)=\begin{cases} 1 & \text{if}\ x= 1/n\ \text{for some}\ n \in \mathbb{N} \\ 0 & \text{otherwise} \end{cases}$$ Show that ...
2
votes
3answers
91 views

Integrate $\int_0^\infty \frac{\sqrt{x}}{e^{(x-\alpha)\beta}+1}dx$

I need to solve for the parameter $\alpha$ after I calculate the integral.$$ \mathcal{R}(\alpha,\beta)=\int_0^\infty \frac{\sqrt{x}}{e^{(x-\alpha)\beta}+1}dx, \ \ \beta >0 $$ The result of this ...
6
votes
1answer
68 views

Integral $I=\int_0^\infty \frac{\ln(1+x) Li_2 (-x)}{x^{3/2}} dx$

Hello can you please help me solve this integral $$ \int_0^\infty \frac{\ln(1+x) Li_2 (-x)}{x^{3/2}} dx=-\frac{2\pi}{3}(\pi^2+24\ln 2). $$ I am trying to work through all logarithmic integrals. Note, ...
7
votes
4answers
161 views

Integrate $ \int_0^\infty \frac{ \ln^2(1+x)}{x^{3/2}} dx=8\pi \ln 2$

I am trying to evaluate this integral. $$ I=\int_0^\infty \frac{ \ln^2(1+x)}{x^{3/2}} dx=8\pi \ln 2 $$ Note $$ \ln(1+x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}x^n}{n}, \ |x| < 1. $$ I was trying to do ...
8
votes
2answers
248 views

Integral $I=\int_0^\infty \frac{\ln(1+x)\ln(1+x^{-2})}{x} dx$

Hi I am stuck on showing that $$ \int_0^\infty \frac{\ln(1+x)\ln(1+x^{-2})}{x} dx=\pi G-\frac{3\zeta(3)}{8} $$ where G is the Catalan constant and $\zeta(3)$ is the Riemann zeta function. Explictly ...
2
votes
1answer
44 views

Integral $6\int_{x=0}^{x=1}\int_{y=x}^{y=1}\int_{z=x}^{z=y} f(x) f(y) f(z)dxdydz=\bigg(\int_0^1 f(t) dt\bigg)^3$

Prove that $$ 6\int_{x=0}^{x=1}\int_{y=x}^{y=1}\int_{z=x}^{z=y} f(x) f(y) f(z)dxdydz=\bigg(\int_0^1 f(t) dt\bigg)^3 $$ assuming $f(x)$ is continuous on [0,1]. This is from an old Putnam exam. I am ...
4
votes
1answer
68 views

Integral $I=\int_0^\infty \frac{e^{\alpha x}-e^{\beta x}}{x(e^{\alpha x}+1)(e^{\beta x}+1)}dx, \ \ \alpha>\beta>0. $

$$ I(\alpha,\beta)=\int_0^\infty \frac{e^{\alpha x}-e^{\beta x}}{x(e^{\alpha x}+1)(e^{\beta x}+1)}dx, \ \ \alpha>\beta>0. $$ I am trying to solve this integral. This is from the old high school ...
12
votes
5answers
306 views

Integral $\lim_{n\to \infty} \int_0^1 \int_0^1…\int_0^1 \cos^2\big(\frac{\pi}{2n}(x_1+x_2+…x_n)\big)dx_1 dx_2…dx_n$

I am trying to evaluate$$ \lim_{n\to \infty} \int_0^1 \int_0^1...\int_0^1 \cos^2\big(\frac{\pi}{2n}(x_1+x_2+...x_n)\big)dx_1 dx_2...dx_n. $$ This is from an old Putnam mathematics competition. Either ...
6
votes
2answers
110 views

Integrating $ \int_2^4 \frac{ \sqrt{\ln(9-x)} }{ \sqrt{\ln(9-x)}+\sqrt{\ln(x+3)} } dx. $

Compute $$ \int_2^4 \frac{ \sqrt{\ln(9-x)} }{ \sqrt{\ln(9-x)}+\sqrt{\ln(x+3)} } dx. $$ I am not sure how to start this one...I am thinking of a substitution to get started.
6
votes
2answers
168 views

Integral $ \int_0^\infty \frac{\ln(1+\sigma x)\ln(1+\omega x^2)}{x^3}dx$

Hello there I am trying to calculate $$ \int_0^\infty \frac{\ln(1+\sigma x)\ln(1+\omega x^2)}{x^3}dx $$ NOT using mathematica, matlab, etc. We are given that $\sigma, \omega$ are complex. Note, the ...
0
votes
1answer
63 views

Improper integral convergence from minus to positive infinity

Quote from Essential Calculus: Early Transcendentals, by James Stewart: If $f$ is continuous, then $$\int_{-\infty}^\infty f(x)dx=\lim_{t \to \infty}\int_{-t}^tf(x)dx$$ I thought this would be ...
1
vote
2answers
69 views

What does Riemann-Stieltjes integral calculate when $\alpha(x) \neq x$?

When we get Riemann-Stieltjes integral becomes standard Riemann integral which calculates area under the curve. We have that $$ s(f,\alpha,P)=\sum_{k=1}^nm_k\Delta\alpha_k \ \text{ and }\ ...
4
votes
1answer
86 views

Integral of composition [duplicate]

Prove that if $f,g:[0,1]\rightarrow[0,1]$ - continuous functions and f is strictly increasing then $$\int\limits_0^1f(g(x))dx\leq\int\limits_0^1f(x)dx+\int\limits_0^1g(x)dx.$$ I tried to prove that ...
16
votes
1answer
495 views

Fourier Transform $J^3_0(x)$.

Hi I am trying to evaluate the integral $$ \mathcal{I}(\omega)=\int_{-\infty}^\infty J^3_0(x) e^{i\omega x}dx $$ analytically. We can also write $$ \mathcal{I}(\omega)=\mathcal{FT}\big(J^3_0(x)\big) ...
0
votes
1answer
30 views

Din derivatives and fundamental theorem of calculus

I have been looking for some references concerning the fundamental theorem of calculus and Dini derivatives and I did not find it. I would like to know if given a locally Lipschitz function ...
0
votes
3answers
101 views

Finding upper and lower bounds for a definite integral

Is there a way to find upper and lower bounds for a definite integral without explicitly finding the antiderivative? For example, $$A\le\int_0^1\frac{1}{\sqrt{1+x^4}}dx\le B$$ I was thinking the mean ...
1
vote
2answers
47 views

Let $f(x)=\int_0^1|t-x|t~dt$ for all real $x$. Sketch the graph of $f(x)$, what is the minimum value of $f(x)$

Let $f(x)=\int_0^1|t-x|t~dt$ for all real $x$. Sketch the graph of $f(x)$, what is the minimum value of $f(x)$ I could not in any way understand how to approach this problem. I think I will be able ...
1
vote
0answers
35 views

Prove an equation about fractional integral

I'm doing this exercise from Real Analysis of Folland and got stuck on this problem. Can anyone help me solve this? I really appreciate. If $f$ is continuous on $[0, \infty)$, for $\alpha \gt ...
4
votes
2answers
125 views

Integral $ \int_0^1 \frac{\ln \ln (1/x)}{1+x^{2p}} dx$…Definite Integral

Calculate $$ I_1:=\int_0^1 \frac{\ln \ln (1/x)}{1+x^{2p}} dx, \ p \geq 1. $$ I am trying to solve this integral $I_1$. I know how to solve a related integral $I_2$ $$ I_2:=\int_0^1 \frac{\ln \ln ...
8
votes
2answers
236 views

Integral…$\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d \theta$.

I am trying to calculate $$ I=\frac{1}{\pi}\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d \theta=\frac{11\pi^4}{180}=\frac{11\zeta(4)}{2}. $$ Note, we can expand the log in the integral to ...