Tagged Questions
2
votes
0answers
43 views
Evaluate $\int_0^\tau \frac{t\sin(t z)}{z\cos(t z)-\sin(tz)}\text{d}t$
I'm trying to evaluate the following definite integral. Mathematica gives me a complicated expression which I think I can simplify, but I was wondering if there was a "nice" way to evaluate it.
...
0
votes
1answer
62 views
A question about the residue calculus
Suppose I have a convergent definite integral of the form $$\int_{-\infty}^\infty \frac{f(x)}{x^2(e^x-1)}\text{d}x,$$
where $f(x)$ has no poles, and I want to try to evaluate it using the residue ...
8
votes
4answers
250 views
Prove $\int_0^\infty \frac{\sin^4x}{x^4}dx = \frac{\pi}{3}$
I need to show that
$$
\int_0^\infty \frac{\sin^4x}{x^4}dx = \frac{\pi}{3}
$$
I have already derived the result $\int_0^\infty \frac{\sin^2x}{x^2} = \frac{\pi}{2}$ using complex analysis, a result ...
13
votes
2answers
275 views
Evaluation of $\int_0^{\pi/3} \ln^2\left(\frac{\sin x }{\sin (x+\pi/3)}\right)\,\mathrm{d}x$
I plan to evaluate
$$\int_0^{\pi/3} \ln^2\left(\frac{\sin x }{\sin (x+\pi/3)}\right)\, \mathrm{d}x$$
and I need a starting point for both real and complex methods. Thanks !
Sis.
4
votes
1answer
135 views
Non elementary antiderivative of $\phi(\cos x,\sin x)$ when $\phi(x,y)$ is a rational real function?
With the method of Residues, we can calculate the integral
\begin{equation}\int_{0}^{2\pi}\phi(\cos x,\sin x)\, dx
\end{equation}
where $\phi(x,y)=\frac{p(x,y)}{q(x,y)}$, ($p,q$ are polynomials of ...
1
vote
3answers
203 views
Evalulate $\int_{-\infty}^{\infty}\frac{1}{(1+x^{2n})^2}dx$ by using residue theorem
I know the answer of the integral $$\int_{-\infty}^{\infty}\frac{1}{1+x^{2n}}dx=\frac{\pi}{n\sin\left(\frac{\pi}{2n}\right)}$$where $n\in\mathbb{N}$.
But how to evalulate ...
6
votes
1answer
655 views
Residue integral: $\int_{- \infty}^{+ \infty} \frac{e^{ax}}{1+e^x} dx$ with $0 \lt a \lt 1$.
I'm self studying complex analysis. I've encountered the following integral:
$$\int_{- \infty}^{+ \infty} \frac{e^{ax}}{1+e^x} dx \text{ with } a \in \mathbb{R},\ 0 \lt a \lt 1. $$
I've done the ...
