Residue computations: $\frac{\sin x}{x}$, Mellin-type, and logarithmic integrals

Compute the following integrals.

1. \(\displaystyle\int_0^\infty \frac{\sin x}{x}\,dx\)

2. \(\displaystyle\int_0^\infty \left(\frac{\sin x}{x}\right)^2 dx\)

3. \(\displaystyle\int_0^\infty \frac{x^{a-1}}{(1+x)^2}\,dx\), \(0 < a < 2\)

4. \(\displaystyle\int_0^\infty \frac{\cos ax - \cos bx}{x^2}\,dx\), \(a, b > 0\)

5. \(\displaystyle\int_0^\infty \frac{x^{a-1}}{1+x^n}\,dx\), \(0 < a < n\)

6. \(\displaystyle\int_0^\infty \frac{\log x}{1+x^n}\,dx\), \(n \ge 2\)

7. \(\displaystyle\int_0^\infty \frac{\log x}{(1+x^2)^2}\,dx\)

8. \(\displaystyle\int_0^\pi \log\abs{1 - a\sin\theta}\,d\theta\), \(a \in \mathbb{C}\)