\(\displaystyle \int_0^\infty\frac{\dx}{x^7+1} = ? \)

(Simplify as much as possible.)

Difficulty: 4. Solution is available for this problem.

Let \(\displaystyle a\in(0,1)\).

\(\displaystyle \int_0^\infty\frac{x^\alpha}{x^2+1}\dx=?\)

Difficulty: 4.

a) \(\displaystyle \displaystyle \int\limits_{|z|=2} \frac{z^{10}}{(z-1)^7}dz=?\)    b) \(\displaystyle \displaystyle \int\limits_{|z|=21} \frac 1{z(z-1) \dots (z-20)}dz=?\)

Difficulty: 4.

\(\displaystyle \int_{|z|=2} \frac{\dz}{(z^4+z^2)\sin z} =? \)

Difficulty: 5.

\(\displaystyle \int_{|z|=2} \frac{\dz}{(z^2+1)\sin z} =? \)

Difficulty: 5.

\(\displaystyle \displaystyle \int\limits_{-\infty}^\infty \frac{x^4-1}{x^6-1}\dx=?\)

Difficulty: 5.

a) \(\displaystyle \displaystyle \int\limits_{C(\pi, 1)} \frac z {\sin z} dz=?\)    b) \(\displaystyle \displaystyle \int\limits_{C(\pi i, 1)} \frac {e^z}{(z-\pi i)^2} dz=?\)

Difficulty: 5.

a) \(\displaystyle \displaystyle \int\limits_{|z-i|=1} \frac{e^{iz}}{1+z^2}dz=?\)  b) \(\displaystyle \displaystyle \int\limits_{|z-\pi|=1} \frac{e^{z}}{\sin^2 z}dz=?\)  c) \(\displaystyle \displaystyle \int\limits_{|z-2\pi i|=1} \frac 1{e^{z}-1}dz=?\)  d) \(\displaystyle \displaystyle \int\limits_{|z|=\pi} \frac{e^{z}}{\cos z-1}dz=?\)

Difficulty: 5.

What residues are possible for \(\displaystyle f'/f\) at \(\displaystyle z_0\) if \(\displaystyle f\) has an isolated singularity in that point?

Difficulty: 5.

\(\displaystyle \frac1{2\pi}\int_{0}^{2\pi}(e^{it}+e^{-it})^ndt=?\)

Difficulty: 5.

\(\displaystyle \int_0^\infty\frac{\cos x}{x^2+1}\dx=? \)

Difficulty: 6.

\(\displaystyle \int_0^\infty\frac{\log x}{x^3+1}\dx=?\)

Difficulty: 6.

\(\displaystyle \int_0^\infty \frac{\log x}{x^2-1} \dx =? \)

Difficulty: 6.

a) \(\displaystyle \displaystyle \int\limits_0^\infty \frac{\cos ax}{x^2+a^2}\dx\)    \(\displaystyle (a>0)\)    b) \(\displaystyle \displaystyle \int\limits_0^\infty \frac{x\sin x}{x^2+a^2}\dx\)

Difficulty: 6.

\(\displaystyle \int_0^\infty\frac{\sqrt{x}}{x^3+1}\dx=? \quad \int_{-\infty}^\infty\frac{e^{-it}}{x^4+1}=? \quad \int_0^\infty\frac{\sin x}x\dx=? \)

Difficulty: 6.

Let \(\displaystyle \Gamma_{r,R,\varepsilon}\) be the curve in the figure, where \(\displaystyle R\) is large, \(\displaystyle r\) is small and \(\displaystyle \varepsilon\) is much smaller than \(\displaystyle r\). What results form the following limit? \(\displaystyle {\displaystyle\lim_{R\to\infty}\lim_{r\to+0}\lim_{\varepsilon\to+0} \frac1{2\pi i}\int_{\Gamma_{r,R,\varepsilon}} \frac{\log z}{z^2+1}\dz}\)

Difficulty: 6.

\(\displaystyle \int_0^\infty\frac\dx{x^3+1}=? \quad \int_0^\infty\frac{\log x}{x^2+x+1}\dx=? \quad \int_0^\infty\frac{\log^2x}{x^2+1}\dx=? \)

Difficulty: 7.

\(\displaystyle \displaystyle \int\limits_{-\infty}^\infty \frac{e^{\alpha t}}{1+e^t}\dt=?\)    \(\displaystyle (0<\alpha<1)\)

Difficulty: 7.

\(\displaystyle \displaystyle \int\limits_{-i\infty}^{i\infty} \frac{\ch Az}{(z+1)(z+2)}dz=?\)    \(\displaystyle (A>0)\)

Difficulty: 7.

\(\displaystyle \int\limits_{-\infty}^\infty \frac{(x-3)\cos x}{x^2-6x+109}\dx=? \)

Difficulty: 7.

Determine for any \(\displaystyle a>0\) the value of the integral \(\displaystyle \displaystyle\frac1{2\pi i}\int\limits_{|z|=2}\frac{a^\xi}{1-\xi^2}\dxi\).

Difficulty: 7.

\(\displaystyle \displaystyle \int\limits_{\sigma -i}^{\sigma +i} \frac{zt^z}{z^2+1}dz=?\)    \(\displaystyle (\sigma>0, \qquad 0<t<1)\)

Difficulty: 7.

Let \(\displaystyle a>0\). Determine

\(\displaystyle \int\limits_{\re z=0}\frac{a^z}{z^2-1}\dz .\)

Difficulty: 7.

a) \(\displaystyle \displaystyle \int\limits_0^\infty \cos x^2 dx=?\)    b) \(\displaystyle \displaystyle \int\limits_{-\infty}^\infty \sin (3x^2+1)dx=?\)

Difficulty: 9.

\(\displaystyle \displaystyle \int\limits_0^{\pi/2} \log\sin x dx=?\)

Difficulty: 9.

Assume that the Dirichlet series \(\displaystyle \displaystyle f(s)=\sum_{n=1}^\infty\frac{a_n}{n^s}\) absolutely converges for \(\displaystyle \re s\ge1\) and let \(\displaystyle X>0\) be real. Find the following integrals:

\(\displaystyle \lim_{h\to\infty}\frac1{2\pi i}\int_{\re s=1, |\im s|\le h}f(z)\frac{X^s}{s}\ds \qquad \frac1{2\pi i}\int_{\re s=1}f(z)\frac{X^s}{s^2}\ds \qquad \frac1{2\pi i}\int_{\re s=1}f(z)\frac{X^s}{s(s+1)}\ds \)

Difficulty: 9.