Appendix A A Short Table of Integrals a. ∫dua2+u2=1aarctan(ua)+C\int \frac{d u}{a^2+u^2}=\frac{1}{a} \arctan \left(\frac{u}{a}\right)+C b. ∫duu2±a2=ln∣u+u2±a2∣+C\int \frac{d u}{\sqrt{u^2 \pm a^2}}=\ln \left|u+\sqrt{u^2 \pm a^2}\right|+C c. ∫u2±a2du=u2u2±a2±a22ln∣u+u2±a2∣+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. ∫u2duu2±a2=u2u2±a2∓a22ln∣u+u2±a2∣+C\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. ∫duuu2+a2=−1aln∣a+u2+a2u∣+C\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. ∫duuu2−a2=1aarcsec(ua)+C\int \frac{d u}{u \sqrt{u^2-a^2}}=\frac{1}{a} \operatorname{arcsec}\left(\frac{u}{a}\right)+C g. ∫dua2−u2=arcsin(ua)+C\int \frac{d u}{\sqrt{a^2-u^2}}=\arcsin \left(\frac{u}{a}\right)+C h. ∫a2−u2du=u2a2−u2+a22arcsin(ua)+C\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. ∫u2a2−u2du=−u2a2−u2+a22arcsin(ua)+C\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. ∫duua2−u2=−1aln∣a+a2−u2u∣+C\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. ∫duu2a2−u2=−a2−u2a2u+C\int \frac{d u}{u^2 \sqrt{a^2-u^2}}=-\frac{\sqrt{a^2-u^2}}{a^2 u}+C
Appendix A A Short Table of Integrals a. ∫dua2+u2=1aarctan(ua)+C\int \frac{d u}{a^2+u^2}=\frac{1}{a} \arctan \left(\frac{u}{a}\right)+C b. ∫duu2±a2=ln∣u+u2±a2∣+C\int \frac{d u}{\sqrt{u^2 \pm a^2}}=\ln \left|u+\sqrt{u^2 \pm a^2}\right|+C c. ∫u2±a2du=u2u2±a2±a22ln∣u+u2±a2∣+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. ∫u2duu2±a2=u2u2±a2∓a22ln∣u+u2±a2∣+C\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. ∫duuu2+a2=−1aln∣a+u2+a2u∣+C\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. ∫duuu2−a2=1aarcsec(ua)+C\int \frac{d u}{u \sqrt{u^2-a^2}}=\frac{1}{a} \operatorname{arcsec}\left(\frac{u}{a}\right)+C g. ∫dua2−u2=arcsin(ua)+C\int \frac{d u}{\sqrt{a^2-u^2}}=\arcsin \left(\frac{u}{a}\right)+C h. ∫a2−u2du=u2a2−u2+a22arcsin(ua)+C\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. ∫u2a2−u2du=−u2a2−u2+a22arcsin(ua)+C\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. ∫duua2−u2=−1aln∣a+a2−u2u∣+C\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. ∫duu2a2−u2=−a2−u2a2u+C\int \frac{d u}{u^2 \sqrt{a^2-u^2}}=-\frac{\sqrt{a^2-u^2}}{a^2 u}+C