# Tagged Questions

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. ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 $$\int_{0}^{2\pi}\phi(\cos x,\sin x)\, dx$$ where $\phi(x,y)=\frac{p(x,y)}{q(x,y)}$, ($p,q$ are polynomials of ...
### 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 ...
### 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 ...