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

$$\int \frac{du}{\sqrt{u^2 \pm a^2}} = \ln \begin{vmatrix}u + \sqrt{u^2 \pm a^2} \end{vmatrix}+ C.$$ (B..9)

$$\int \sqrt{u^2 \pm a^2} \; du = \frac u2 \; \sqrt{u^2 \pm a^2} \pm \frac{a^2}{2} \ln \begin{vmatrix}u + \sqrt{u^2 \pm a^2} \end{vmatrix}+C.$$ (B..10)

$$\int u^2\; \sqrt{u^2 \pm a^2} \; du = \frac u8 (2u^2 \pm a 2 )\; \sqrt{u^2 \pm a^2} -\frac{a^4}{8} \ln \begin{vmatrix}u +\sqrt{u^2 \pm a^2} \end{vmatrix} + C.$$ (B..11)

$$\int \frac{\sqrt{u^2 + a^2}}{u} \; du =\sqrt{u^2 + a^2} - a \ln \begin{vmatrix}a + \sqrt{u^2 + a^2}{u} \end{vmatrix} + C.$$ (B..12)

$$\int \frac{\sqrt{u^2 - a^2}}{u} \; du = \sqrt{u^2 - a^2} - a \sec^{- 1} \begin{vmatrix}\frac ua \end{vmatrix} + C.$$ (B..13)

$$\int \frac {\sqrt{u^2 \pm a^2}}{u^2} \; du = -\frac{ \sqrt{u^2 \pm a^2}}{u} + \ln \begin{vmatrix}u+ \sqrt{u^2 \pm a^2} \end{vmatrix} + C.$$ (B..14)

$$\int \frac{u^2}{\sqrt{u^2 \pm a^2}} \; du = \frac u2 \; \sqrt{u^2 \pm a^2} - \frac {\pm a^2}{2} \ln u + \begin{vmatrix}u+\sqrt{u^2 \pm a^2} \end{vmatrix}+ C.$$ (B..15)

$$\int \frac{du}{u \sqrt{u^2 + a^2}} = -\frac 1a \ln \begin{vmatrix}\frac {a + \sqrt{u^2 + a^2}}{u} \end{vmatrix}+ C.$$ (B..16)

$$\int \frac {du}{ u \sqrt{u^2- a^2}} = \frac 1a \sec^{- 1} \begin{vmatrix}\frac ua \end{vmatrix}+ C.$$ (B..17)

$$\int \frac{du}{u^2 \; \sqrt{u^2 \pm a^2}} = -\sqrt{u^2 \pm a^2} \pm a^2 u + C.$$ (B..18)

$$\int (u^2 \pm a^2 )^{3/2} \; du = \frac u8 (2u^2 \pm 5a^2 )\sqrt{ u^2 \pm a^2} + \frac{3a^4}{8} \ln \begin{vmatrix}u + \sqrt{u^2 \pm a^2} \end{vmatrix}+ C.$$ (B..19)

$$\int \frac{du}{(u^2 \pm a^2 )^{3/2}} = \frac{u}{\pm a^2 \; \sqrt{u^2 \pm a^2}} + C.$$ (B..20)