5
votes
2answers
92 views

Integrate $I=\int_0^1\frac{\ln x}{x^n-1}dx$

Hi I am trying to obtain a closed form for$$ I_n=\int_0^1\frac{\ln x}{x^n-1}dx, \quad n\geq 1. $$ This integral is quite nice and generates many other known closed form results such as $$ ...
1
vote
0answers
28 views

Integral $I=\int_0^1 \frac{\arctan\big(\sqrt{x^2 + 2}\big)}{\sqrt{x^2 + 2}(x^2 + 1)}dx$

Hi I'm trying to show that $$ I=\int_0^1 \frac{\arctan\big(\sqrt{x^2 + 2}\big)}{\sqrt{x^2 + 2}(x^2 + 1)}dx=\frac{5\pi^2}{96}. $$ We can try the substitution $u=(x^2+2)^{1/2}, du=x(2+x^2)^{-1/2}dx$ ...
0
votes
0answers
41 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 ...
1
vote
0answers
31 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
109 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 $\pi/3$, 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
1answer
74 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
109 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
2answers
122 views

Cool Integral $\int_0^{\pi/2}dx\ln \sinh x$

$$ I_1=\int_0^{\pi/2}dx\ln \sinh x,\quad I_2=\int_0^{\pi/2}dx\ln \cosh x, \quad I_1\neq I_2. $$ I am trying to calculate these integrals. We know the similar looking integrals $$ \int_0^{\pi/2}dx\ln ...
2
votes
0answers
56 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
28 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
0answers
46 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
125 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
48 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, ...
3
votes
2answers
94 views

Integral $ \int_{-\pi/2}^{\pi/2} \frac{1}{2007^x+1}\cdot \frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx $

I am trying to solve this integral $$ \int_{-\pi/2}^{\pi/2} \frac{1}{2007^x+1}\cdot \frac{\sin^{2008}x}{\sin^{2008}x+\cos^{2008}x}dx $$ A closed form does exist despite the looks of the integrand. ...
1
vote
1answer
30 views

Hilbert Transform of cos wt = sinwt.

Hilbert Transform of cos wt = sin wt. Can anyone help me with the proof. in Last Step how this become pi
4
votes
0answers
74 views

Ugly-nice double series

I'm trying to evaluate the following ugly double sum (presented in raw notation as used in my calculations): $\sum _{m=1}^{\infty } \sum _{n=1}^{\infty } \frac{4 m \cos \left(\frac{2 \pi m ...
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 ...
7
votes
4answers
158 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 ...
3
votes
2answers
72 views

erf(a+ib) error function separate into real and imaginary part

Is there an easy way to separate erf(a+ib) into real and imaginary part?
3
votes
2answers
121 views

Integral $\int_0^a \ln \left( \frac{b-\sqrt{a^2-x^2}}{b+\sqrt{a^2-x^2}} \right)dx$

Hi I am trying to calculate, $$ \int_0^a \ln \left( \frac{b-\sqrt{a^2-x^2}}{b+\sqrt{a^2-x^2}} \right)dx $$ where $a,b$ are positive real constants. I Know $\ln(xy)=\ln x +\ln y$, but I do not ...
1
vote
2answers
63 views

Contour Integral for Cosine and a rational function

I've been trying to figure out this integral via use of residues: $$\int_{-\infty}^{\infty} \displaystyle \frac{\cos{5x}}{x^4+1}dx$$ The usual semicircle contour wont work for this guy as the ...
3
votes
2answers
109 views

Trouble with $\int_0^\infty e^{-ix^2}\mathrm{d}x$

I'm trying to evaluate $$ \int_0^\infty \mathrm{d}x\ e^{-ix^2}. $$ I tried to integrate on the following contour $\Gamma_R$: the frontier of a circular sector, centered at the origin, of angle $\pi / ...
3
votes
1answer
131 views

Laplace transform of and impulse sampled function using “frequency” convolution

This is a long question, but assume we have this: The book uses the frequency convolution theorem to solve this problem. To solve the integral, it uses a contour + residue theorem to solve it. The ...
0
votes
1answer
38 views

Computing $\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t$

I'd like to calculate the following integral on the interval $[0,2\pi]$: $$ I=\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t = 2\pi. $$
6
votes
2answers
81 views

calculation of $\int^{\frac{\pi}{2}}_{0}\cos^{n}x\cos (nx)\ dx $, where $n\in \mathbb{N}$

Calculation of $\displaystyle \int^{\frac{\pi}{2}}_{0}\cos^{n}x\cos (nx)\ dx $, where $n\in \mathbb{N}$ $\bf{My\; Try}::$ Using $\displaystyle \cos (x) = \frac{e^{ix}+e^{-ix}}{2}$, we get ...
5
votes
1answer
115 views

Prove $\sin a=\int_{-\infty}^{\infty}\cos(ax^2)\frac{\sinh(2ax)}{\sinh(\pi x)} \operatorname dx$

Derive the integral representation $$\sin a=\int_{-\infty}^{\infty}\cos(ax^2)\frac{\sinh(2ax)}{\sinh(\pi x)}dx$$ for $|a|\le \pi/2$.
0
votes
1answer
54 views

Work done by gravitational force

In my calculus class we learned about line integrals, and for homework we have exercise to find work done by gravitational force on material dot with mass $m$ which follows path of the elipse ...
3
votes
1answer
39 views

Integration of exponential with square

It is known that $\int_\mathbb{R}e^{-tx^2}dx=\sqrt{\pi/t}$. What about $\int_\mathbb{R}e^{-t(x+ai)^2}dx$ for $a\in\mathbb{R}$? Is it still also $\sqrt{\pi/t}$? I can't simply change the variable ...
119
votes
5answers
29k views

Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \ \mathrm dx$

I need help with this integral: $$I=\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right)\ \mathrm dx.$$ The integrand graph looks like this: $\hspace{1in}$ The ...
14
votes
1answer
294 views

Further our knowledge of a certain class of integral involving logarithms.

$\newcommand{\limitp}{\alpha}\newcommand{\innerp}{\beta}$I am fascinated by definite integrals. Exploring math.stackexchange, I have found many interesting integrals of the form $$ ...
0
votes
3answers
114 views

Evaluate $\oint_C\frac{dz}{z-2}$ around the square with vertices $3 \pm 3i, -3 \pm 3i$.

I'm having a tough time figuring out $\gamma(t)$ in $\oint_C f(z) dz = \int_a^b f(\gamma(t)) \gamma '(t) dt$. I think I have to split this into four parts, the four separate lines, but from there I'm ...
6
votes
2answers
673 views

How prove this integral $\int_0^{2\pi }{\frac{{{e^{\cos x}}\cos\left({\sin x}\right)}}{{p-\cos\left({y-x}\right)}}}dx$

show that $$ \int\limits_0^{2\pi }{\frac{{{e^{\cos x}}\cos\left({\sin x}\right)}}{{p-\cos\left({y-x}\right)}}}dx =\frac{{2\pi }}{{\sqrt{{p^2}-1}}}\exp\left({\frac{{\cos ...
13
votes
3answers
777 views

How do I evaluate this integral $\int\limits_0^\pi{\frac{{{x^2}}}{{\sqrt 5-2\cos x}}}\operatorname d\!x$?

Show that $$\int\limits_0^\pi{\frac{{{x^2}}}{{\sqrt 5-2\cos x}}}\operatorname d\!x =\frac{{{\pi^3}}}{{15}}+2\pi{\ln^2}\left({\frac{{1+\sqrt 5 }}{2}}\right).$$ I don't have any idea how to start, ...
8
votes
1answer
431 views

Another challenging integral

$$\int_0^{\infty}\frac{e^{-a x^2(x^2-\pi^2)}\cos(2\pi a x^3)}{\cosh x}dx=\frac{\pi}{2}e^{-\pi^4 a/16}.$$ Note the unusual appearance of $x^1,x^2,x^3,x^4$.
5
votes
1answer
305 views

Inverse Laplace Transform of $\bar p_D = \frac{K_0(\sqrt[]s r_D)}{sK_0(\sqrt[]s)}$

I solved the following partial differential equation using Laplace Transform: $\LARGE \frac{1}{r_D}\frac{\partial}{\partial r_D}(r_D\frac{\partial p_D}{\partial r_D})=\frac{\partial p_D}{\partial ...
1
vote
1answer
52 views

Evaluating a contour integral

Find the contour integral of $$\int_C\frac{(z+a)(z+b)}{(z-a)(z-b)} \mbox{d}z,$$ where the modulus of $a$ and $b$ are less than $1$, and the integral path $C$ is the anticlockwise unit circle ($|z|= ...
7
votes
2answers
317 views

Integrating $\int_0^\infty \frac{\log x}{(1+x)^3}\,dx$ using residues

I am trying to use residues to compute $$\int_0^\infty\frac{\log x}{(1+x)^3}\,dx.$$My first attempt involved trying to take a circular contour with the branch cut being the positive real axis, but ...
8
votes
1answer
619 views

Integral of $e^{-x^2}\cos(x^2)$ using residues

I want to solve the following integral: $$\int_0^{\infty} \!\! \operatorname{e}^{-x^2}\!\cos(x^2) \, \operatorname{d}\!x$$ I have seen this in a section about residues, so my guess is that I would ...
4
votes
3answers
259 views

show that $\int_{0}^{\infty}\frac{x\cos ax}{\sinh x}dx=\frac{\pi^2}{4} \operatorname{sech}^2 \left(\frac{a\pi}{2}\right) $

show that $$\int_{0}^{\infty}\frac{x\cos ax}{\sinh x}dx=\frac{\pi^2}{4} \operatorname{sech}^2\left(\frac{a\pi}{2}\right) $$ also I think we can solve it by contour integration but how and its better ...
2
votes
1answer
69 views

show that $\int_{-\infty}^{\infty} \frac {(\sin x) (x^2+a^2)}{x(x^2+b^2)}dx=\frac{\pi(a^2+e^{-b}(b^2-a^2))}{b^2}$

show that $$\int_{-\infty}^{\infty} \frac {(\sin x) (x^2+a^2)}{x(x^2+b^2)}dx=\frac{\pi(a^2+e^{-b}(b^2-a^2))}{b^2}$$ for every $a,b>0$ thanks for all
5
votes
5answers
478 views

show that $\int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8}$

show that $$\int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8}$$ using different ways thanks for all
4
votes
1answer
201 views

integral involving cot coth and e using contour

I am thinking it is possible to evaluate this integral using contours, but I got hung up. $\displaystyle \int_{0}^{\infty}e^{-ax}\sin(ax)\left(\cot(x)+\coth(x)\right)dx=\frac{\pi}{2}\cdot ...
0
votes
2answers
55 views

Power series in complex analysis

We derived from cauchy's integral formula that a holomorphic function converges locally in a power series. Now we had the Identity theorem and I wanted to know whether I can conclude from this that ...
2
votes
1answer
75 views

Complex Analysis: Are these properties equivalent?

I am somehow thinking that these properties must be equivalent, unfortunately I do not know a theorem that says it: $f$ has a global antiderivative iff the line integral $ \int_{\gamma}f$ over every ...
2
votes
1answer
86 views

Inequality contour integral

Given a curve $\gamma:[- \frac{\pi}{6}, \frac{\pi}{6}] \rightarrow \mathbb{C}$ and $\mathop{\gamma}\,(t):=2\exp(it)$, I want to show that $$\left\vert\,\int_{\gamma} \frac{1}{z^3+1} \mathrm{d}z\, ...
4
votes
1answer
59 views

Complex integral, correct?

I am supposed to do the integral $$ \int_{\gamma_2} \frac{\sin(z)}{z+\frac{i}{2}} dz$$ where $\gamma_2:[-\pi, 3\pi] \rightarrow \mathbb{C}$ , $\gamma_2(t)=\exp(it)$ for $ t\in [-\pi,\pi]$, ...
2
votes
1answer
117 views

Simple conceptual Fundamental Thm of Calculus question

When applying the Fundamental Thm of Calculus in complex analysis, what does it mean for an open connected set to contain a loop? For example, does my red-color open annulus contain the black colour ...
1
vote
1answer
106 views

Evaluating the following integral:

I am trying to evaluate this integral: $$\int_{0}^{\infty }\frac{\cos(x)}{1+x^{2}}dx$$ My attempt: $$\int_0^{\infty}\frac{\cos(x)}{(x+i)(x-i)}dx=1/2 \int_{-\infty}^{\infty} ...
1
vote
1answer
179 views

Mellin transform for sin x

I am trying to find the Mellin transform for $\sin x $, in other words $\int^{\infty}_0 (\sin x) x^{s-1} \mathrm{d} x $ and I know that the answer is $\Gamma(s) \sin (\pi s/2)$ from several tables ...
0
votes
4answers
125 views

Contour Integral help with residue theorem

$$ K = \int_{0}^{\infty}\frac{1}{x^{4}+x^{2}+1}dx $$ I am supposed to use contour integration to solve this, but I can't even determine the singularities. The denominator doesn't have any that I can ...