Math 6420 • PDE

Integrate \[ \begin {array}{rcl} \text {(a)}&\displaystyle \int _0^{2\pi }\frac {d\phi }{(p+q\cos \phi )^2} &\text {(parameters }p>q>0\text {)},\\[2.2ex] \text {(b)}&\displaystyle \int _{-\infty }^{\infty }\frac {e^{\alpha x}}{1+e^x}\,dx &\text {(parameter }\alpha \in (0,1)\text {)},\\[2.2ex] \text {(c)}&\displaystyle \int _0^{\infty }\sin (a^2x^2)\,dx &\text {(positive parameter }a\text {)}. \end {array} \]