Compute the following integrals.
\(\displaystyle\int_0^\infty \frac{1}{(1+x^n)^2}\,dx\), \(n \ge 1\)
\(\displaystyle\int_0^\infty \frac{\cos x}{(x^2+a^2)^2}\,dx\), \(a \in \mathbb{R}\)
\(\displaystyle\int_0^\pi \frac{1}{a + \sin\theta}\,d\theta\), \(a > 1\)
\(\displaystyle\int_0^{\frac{\pi}{2}} \frac{d\theta}{a + \sin^2\theta}\), \(a > 0\).
\(\displaystyle\int_{\abs{z}=2} \frac{1}{(z^5-1)(z-3)}\,dz\)
\(\displaystyle\int_{-\infty}^\infty \frac{\sin \pi a}{\cosh \pi x + \cos \pi a}\,e^{-ix\xi}\,dx\), \(0 < a < 1\), \(\xi \in \mathbb{R}\)
\(\displaystyle\int_{\abs{z}=1} \cot^2 z\,dz\).
The source labels these integrals (i), (ii), (iii), (iv), (v), (v), (vi); here they are numbered consecutively, so the source’s second (v) is (vi) and its (vi) is (vii).
Erratum: integral (ii) diverges for \(a = 0\), where the integrand is \(\cos x / x^4\), which is not integrable at \(0\). The hypothesis should be \(a \in \mathbb{R}\), \(a \ne 0\).