TBIL-Calc A Short Table of Integrals

Appendix A A Short Table of Integrals

$\int \frac{d u}{a^{2} + u^{2}} = \frac{1}{a} \arctan (\frac{u}{a}) + C$
$\int \frac{d u}{\sqrt{u^{2} \pm a^{2}}} = \ln | u + \sqrt{u^{2} \pm a^{2}} | + C$
$\int \sqrt{u^{2} \pm a^{2}} d u = \frac{u}{2} \sqrt{u^{2} \pm a^{2}} \pm \frac{a^{2}}{2} \ln | u + \sqrt{u^{2} \pm a^{2}} | + C$
$\int \frac{u^{2} d u}{\sqrt{u^{2} \pm a^{2}}} = \frac{u}{2} \sqrt{u^{2} \pm a^{2}} \mp \frac{a^{2}}{2} \ln | u + \sqrt{u^{2} \pm a^{2}} | + C$
$\int \frac{d u}{u \sqrt{u^{2} + a^{2}}} = - \frac{1}{a} \ln | \frac{a + \sqrt{u^{2} + a^{2}}}{u} | + C$
$\int \frac{d u}{u \sqrt{u^{2} - a^{2}}} = \frac{1}{a} arcsec (\frac{u}{a}) + C$
$\int \frac{d u}{\sqrt{a^{2} - u^{2}}} = \arcsin (\frac{u}{a}) + C$
$\int \sqrt{a^{2} - u^{2}} d u = \frac{u}{2} \sqrt{a^{2} - u^{2}} + \frac{a^{2}}{2} \arcsin (\frac{u}{a}) + C$
$\int \frac{u^{2}}{\sqrt{a^{2} - u^{2}}} d u = - \frac{u}{2} \sqrt{a^{2} - u^{2}} + \frac{a^{2}}{2} \arcsin (\frac{u}{a}) + C$
$\int \frac{d u}{u \sqrt{a^{2} - u^{2}}} = - \frac{1}{a} \ln | \frac{a + \sqrt{a^{2} - u^{2}}}{u} | + C$
$\int \frac{d u}{u^{2} \sqrt{a^{2} - u^{2}}} = - \frac{\sqrt{a^{2} - u^{2}}}{a^{2} u} + C$