Linked Questions

3
votes
3answers
890 views

Integrating $\frac{1}{1+z^3}$ over a wedge to compute $\int_0^\infty \frac{dx}{1+x^3}$. [duplicate]

Compute $\displaystyle\int_0^\infty \frac{dx}{1+x^3}$ by integrating $\dfrac{1}{1+z^3}$ over the contour $\gamma$ (defined below) and letting $R\rightarrow \infty$. The contour is $\gamma=\...
2
votes
1answer
415 views

How to prove this integral problem: $\int_0^{\infty}\frac{dx}{1+x^n}=\frac{\pi}{n}\csc\frac \pi n$? [duplicate]

I dont know if this has already been asked. How to prove this integral $$\int_0^{\infty}\frac{dx}{1+x^n}=\frac{\pi}{n}\csc\frac \pi n\ {?}$$ $n\ge 2$ is a positive integer Frankly speaking i have ...
3
votes
9answers
281 views

Other ways to evaluate the integral $\int_{-\infty}^{\infty} \frac{1}{1+x^{2}} \, dx$?

$$\int_{-\infty}^{\infty}\frac{1}{x^2+1}\,dx=\pi $$ I can do it with the substitution $x= \tan u$ or complex analysis. Are there any other ways to evaluate this?
33
votes
1answer
619 views

Interpretation of an equality: Area of regular polygon and the integral $\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^N}$

Recently, I learned the integral from this post: $$\mathcal I=\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^N}=\frac{\pi/N}{\sin(\pi/N)}$$ This reminds me of the area of a regular polygon. Consider a $2N$...
1
vote
4answers
295 views

How to integrate $\int_0^\infty \frac{1}{1+y^4} dy$ [duplicate]

I tried the trigonometric substitution $y^2 = \tan \theta, sec^2\theta = 1 + y^4$ But now I'm stuck with $\frac12 \int \frac{\sqrt{\sin \theta}}{(\cos\theta)^{\frac92} } d \theta$ I ran out of ...
2
votes
3answers
112 views

Calculate :$\int_{0}^{\infty}{ }\frac{dx}{a^6+(x-\frac{1}{x})^6}$

Find the: $$\int_{0}^{\infty}{ }\frac{dx}{a^6+(x-\frac{1}{x})^6}:a>0$$ My Try: $$\frac{1}{a^6}\int_{0}^{\infty}{ }\frac{dx}{1+(\frac{x-\frac{1}{x}}{a})^6}$$ $$\int_0^\infty{dx\over1+u^6}={1\...
1
vote
3answers
280 views

Show that $\displaystyle{\int_{0}^{\infty}\!\frac{x^{a}}{x(x+1)}~\mathrm{d}x=\frac{\pi}{\sin(\pi a)}}$

Show that for $0<a<1$ $$\int_{0}^{\infty}\frac{x^{a}}{x(x+1)}~\mathrm{d}x=\frac{\pi}{\sin(\pi a)}$$ I want to solve this question by using complex analysis tools but I even don't know how to ...
1
vote
2answers
160 views

Why should I take this Contour for $I = \int_0^{\infty} \frac{dx}{1+x^3} $? (Analytic Continuation)

When discussing analytic continuation, my lecturer used the following example, $$ I = \int_0^{\infty} \frac{dx}{1+x^3} $$ I have in my notes that the contour was taken as below. I must admit I was ...
7
votes
0answers
548 views

How to integrate $\int_0^\infty\frac{dx}{1+x^n}$ [duplicate]

I was playing around with the function $\dfrac{1}{1+x^2}$, and knowing that the integral over $(0,\infty)$ was $\dfrac{\pi}{2}$, I was hoping to see if there was some neat pattern to determining the ...
1
vote
1answer
301 views

Using Residue theorem to evaluate the integral

Using Residue theorem to evaluate the integral: $$\int_0^{\infty} \frac{x^2}{x^4 + 5x^2 +6}dx$$ I am using partial fraction to get: $$\int_0^{\infty} \left( \frac{3}{x^2 +3} - \frac{2}{x^2+2} \...
0
votes
3answers
155 views

Solution for the Laplace Transform $\mathscr{L}\left\{\frac{t^{\alpha-1}}{t-\mu}\right\}(\beta)$

I have been looking for an explicit solution to the following Laplace transform for $\alpha,\mu,\beta>0$ \begin{equation} \frac{\beta^\alpha}{\Gamma(\alpha)}\mathscr{L}\left\{\frac{t^{\alpha-1}}{t-\...
1
vote
2answers
118 views

Evaluate the integral $\int_0^{\infty} \frac{dx}{x^{\frac{\alpha}{2}}+1}$ for the general $\alpha$

I want to find the solution to the following integral $$\int_0^{\infty} \frac{dx}{x^{\frac{\alpha}{2}}+1}$$ where $\alpha$ can be any value greater than 2 such that $\alpha /2 >1$ but can be any ...
3
votes
1answer
83 views

Residue theorem for line segment

I am working through this problem:- Show that $\int_0^ \infty \frac{1}{1+x^n} dx= \frac{ \pi /n}{\sin(\pi /n)}$ , where $n$ is a positive integer. I follow it all, except for part (3) - I think this ...
1
vote
0answers
77 views

Why $\lim\limits_{R \to +\infty} \int_{[0,Re^{i2\pi/n}]}\frac{1}{1+z^n}dz$ = $e^{i2\pi/n}$ $\int_0^{+\infty} \frac{1}{1+x^n}dx$

Why $\lim\limits_{R \to +\infty} \int_{[0,Re^{i2\pi/n}]}\frac{1}{1+z^n}dz$ = $e^{i2\pi/n}$ $\int_0^{+\infty} \frac{1}{1+x^n}dx$ (The integration is on the ray: ${[0,Re^{i2\pi/n}]}$) As it is shown ...
1
vote
2answers
28 views

Residue integral why $-e^{2\pi i/n}$ term comes for inclined line?

$$-e^{2\pi i/n}\int_0^\infty\frac1{1+x^n}\mathrm{d}x\tag{3}$$ from Show that $\int_0^ \infty \frac{1}{1+x^n} dx= \frac{ \pi /n}{\sin(\pi /n)}$ , where $n$ is a positive integer. this link I dont ...

15 30 50 per page