# Active Calculus

## AppendixAA Short Table of Integrals

1. $$\displaystyle \int \frac{du}{a^2 + u^2} = \frac{1}{a} \arctan \left( \frac{u}{a} \right) + C$$
2. $$\displaystyle \int \frac{du}{\sqrt{u^2 \pm a^2}} = \ln |u + \sqrt{u^2 \pm a^2}| + C$$
3. $$\displaystyle \int \sqrt{u^2 \pm a^2} \, du = \frac{u}{2}\sqrt{u^2 \pm a^2} \pm \frac{a^2}{2}\ln|u + \sqrt{u^2 \pm a^2}| + C$$
4. $$\displaystyle \int \frac{u^2 du}{\sqrt{u^2 \pm a^2}} = \frac{u}{2}\sqrt{u^2 \pm a^2} \mp \frac{a^2}{2}\ln|u + \sqrt{u^2 \pm a^2}| + C$$
5. $$\displaystyle \int \frac{du}{u\sqrt{u^2+a^2}} = -\frac{1}{a} \ln \left| \frac{a+\sqrt{u^2 + a^2}}{u} \right| + C$$
6. $$\displaystyle \int \frac{du}{u\sqrt{u^2-a^2}} = \frac{1}{a} \arcsec \left( \frac{u}{a} \right) + C$$
7. $$\displaystyle \int \frac{du}{\sqrt{a^2 - u^2}} = \arcsin \left( \frac{u}{a} \right) + C$$
8. $$\displaystyle \int \sqrt{a^2 - u^2} \, du = \frac{u}{2}\sqrt{a^2 - u^2} + \frac{a^2}{2} \arcsin \left( \frac{u}{a} \right) + C$$
9. $$\displaystyle \int \frac{u^2}{\sqrt{a^2 - u^2}} \, du = -\frac{u}{2}\sqrt{a^2 - u^2} + \frac{a^2}{2} \arcsin \left( \frac{u}{a} \right) + C$$
10. $$\displaystyle \int \frac{du}{u\sqrt{a^2 - u^2}} = -\frac{1}{a} \ln \left| \frac{a + \sqrt{a^2 - u^2}}{u} \right| + C$$
11. $$\displaystyle \int \frac{du}{u^2\sqrt{a^2 - u^2}} = -\frac{\sqrt{a^2-u^2}}{a^2 u} + C$$