The primary focus of this work is the closed-form evaluation of integrals of the form$$\int\sbsp{0}{\infty}{P(x\sp2)\over Q(x\sp2)} dx,$$where P and Q are polynomials in x Most of our results are expressed in terms of the polynomial $$\eqalign{\Phi(a\sb1,\cdots,a\sb{p},q;x)&=x\sp{q}+a\sb{p}x \sp{q-1}+a\sb{p-1}x\sp{q-2}\cr\cdots{+}a\sb1x\sp{q/2}{+}a\sb2x\sp{q/2{-}1}{+}\cdots{+}1,\cr}$$and the integrals$$M\sb{q}(a\sb1,\cdots,a\sb{p};m)=\int\sbsp{0}{\infty} \left\lbrack{x\sp{q/2}\over \Phi(a\sb1,\cdots,a\sb{p},q;x)} \right\rbrack\sp{m}\ dx$$and$$ N\sb{j,q}(a\sb1,\cdots,a\sb{p};m)=\int\sbsp{0}{\infty} \left\lbrack{x\sp{2j}\over \Phi(a\sb1,\cdots,a\sb{p},q;x)}\right\rbrack\sp{m+1}\ dx,$$where $q=4p$ or $4p+2$ according to the residue of q modulo 4 A number of other types of results follow from our integral evaluations. These include some series, among the most notable being$$\eqalign{\sqrt{a+\sqrt{1+c}}\ =\ &\sqrt{a+1}+\cr &+{1\over \pi\sqrt{2}}\sum\sbsp{k=0}{\infty}{(-1)\sp{k-1}\over k}N\sb{0,4}(a;k-1).\cr}$$ The special case with $a=1$ has been studied extensively by Lagrange, Ramanujan and others