Appendix A A Short Table of Integrals
a. \[\int \frac{d u}{a^2+u^2}=\frac{1}{a} \arctan \left(\frac{u}{a}\right)+C\]
b. \[\int \frac{d u}{\sqrt{u^2 \pm a^2}}=\ln \left|u+\sqrt{u^2 \pm a^2}\right|+C\]
c. \[\int \sqrt{u^2 \pm a^2} d u=\frac{u}{2} \sqrt{u^2 \pm a^2} \pm \frac{a^2}{2} \ln \left|u+\sqrt{u^2 \pm a^2}\right|+C\]
d. \[\int \frac{u^2 d u}{\sqrt{u^2 \pm a^2}}=\frac{u}{2} \sqrt{u^2 \pm a^2} \mp \frac{a^2}{2} \ln \left|u+\sqrt{u^2 \pm a^2}\right|+C\]
e. \[\int \frac{d u}{u \sqrt{u^2+a^2}}=-\frac{1}{a} \ln \left|\frac{a+\sqrt{u^2+a^2}}{u}\right|+C\]
f. \[\int \frac{d u}{u \sqrt{u^2-a^2}}=\frac{1}{a} \operatorname{arcsec}\left(\frac{u}{a}\right)+C\]
g. \[\int \frac{d u}{\sqrt{a^2-u^2}}=\arcsin \left(\frac{u}{a}\right)+C\]
h. \[\int \sqrt{a^2-u^2} d u=\frac{u}{2} \sqrt{a^2-u^2}+\frac{a^2}{2} \arcsin \left(\frac{u}{a}\right)+C\]
i. \[\int \frac{u^2}{\sqrt{a^2-u^2}} d u=-\frac{u}{2} \sqrt{a^2-u^2}+\frac{a^2}{2} \arcsin \left(\frac{u}{a}\right)+C\]
j. \[\int \frac{d u}{u \sqrt{a^2-u^2}}=-\frac{1}{a} \ln \left|\frac{a+\sqrt{a^2-u^2}}{u}\right|+C\]
k. \[\int \frac{d u}{u^2 \sqrt{a^2-u^2}}=-\frac{\sqrt{a^2-u^2}}{a^2 u}+C\]