Integrals containing $ \sqrt {u^2 \pm a^2}$ .

$\displaystyle \int \frac{du}{\sqrt{u^2 \pm a^2}}$ $\displaystyle = \ln \begin{vmatrix}u + \sqrt{u^2 \pm a^2} \end{vmatrix}+ C.$ (B..9)
$\displaystyle \int \sqrt{u^2 \pm a^2} \; du$ $\displaystyle = \frac u2 \; \sqrt{u^2 \pm a^2} \pm \frac{a^2}{2} \ln \begin{vmatrix}u + \sqrt{u^2 \pm a^2} \end{vmatrix}+C.$ (B..10)
$\displaystyle \int u^2\; \sqrt{u^2 \pm a^2} \; du$ $\displaystyle = \frac u8 (2u^2 \pm a 2 )\; \sqrt{u^2 \pm a^2} -\frac{a^4}{8} \ln \begin{vmatrix}u +\sqrt{u^2 \pm a^2} \end{vmatrix} + C.$ (B..11)
$\displaystyle \int \frac{\sqrt{u^2 + a^2}}{u} \; du$ $\displaystyle =\sqrt{u^2 + a^2} - a \ln \begin{vmatrix}a + \sqrt{u^2 + a^2}{u} \end{vmatrix} + C.$ (B..12)
$\displaystyle \int \frac{\sqrt{u^2 - a^2}}{u} \; du$ $\displaystyle = \sqrt{u^2 - a^2} - a \sec^{- 1} \begin{vmatrix}\frac ua \end{vmatrix} + C.$ (B..13)
$\displaystyle \int \frac {\sqrt{u^2 \pm a^2}}{u^2} \; du$ $\displaystyle = -\frac{ \sqrt{u^2 \pm a^2}}{u} + \ln \begin{vmatrix}u+ \sqrt{u^2 \pm a^2} \end{vmatrix} + C.$ (B..14)
$\displaystyle \int \frac{u^2}{\sqrt{u^2 \pm a^2}} \; du$ $\displaystyle = \frac u2 \; \sqrt{u^2 \pm a^2} - \frac {\pm a^2}{2} \ln u + \begin{vmatrix}u+\sqrt{u^2 \pm a^2} \end{vmatrix}+ C.$ (B..15)
$\displaystyle \int \frac{du}{u \sqrt{u^2 + a^2}}$ $\displaystyle = -\frac 1a \ln \begin{vmatrix}\frac {a + \sqrt{u^2 + a^2}}{u} \end{vmatrix}+ C.$ (B..16)
$\displaystyle \int \frac {du}{ u \sqrt{u^2- a^2}}$ $\displaystyle = \frac 1a \sec^{- 1} \begin{vmatrix}\frac ua \end{vmatrix}+ C.$ (B..17)
$\displaystyle \int \frac{du}{u^2 \; \sqrt{u^2 \pm a^2}}$ $\displaystyle = -\sqrt{u^2 \pm a^2} \pm a^2 u + C.$ (B..18)
$\displaystyle \int (u^2 \pm a^2 )^{3/2} \; du$ $\displaystyle = \frac u8 (2u^2 \pm 5a^2 )\sqrt{ u^2 \pm a^2} + \frac{3a^4}{8} \ln \begin{vmatrix}u + \sqrt{u^2 \pm a^2} \end{vmatrix}+ C.$ (B..19)
$\displaystyle \int \frac{du}{(u^2 \pm a^2 )^{3/2}}$ $\displaystyle = \frac{u}{\pm a^2 \; \sqrt{u^2 \pm a^2}} + C.$ (B..20)

Noah Dana-Picard 2007-12-28