APPM 1360 -- Exam #1 Formula Sheet -- February 21, 2001

A short table of integrals. In the following, $a \neq 0$ .

1.: $\displaystyle\int \frac{du}{\sqrt{a^2 - u^2}} = \sin^{-1}(u/a) + C$ for u² < a²
2.: $\displaystyle\int \frac{du}{a^2+u^2} = (1/a) \tan^{-1}(u/a) + C$
3.: $\displaystyle\int \frac{du}{u \sqrt{u^2-a^2}} = (1/a) \sec^{-1} \vert u/a\vert + C$ for u² > a² +C
4.: $\displaystyle\int \frac{du}{\sqrt{a^2 + u^2}} = \sinh^{-1} (u/a) + C$ for a>0
5.: $\displaystyle\int \frac{du}{\sqrt{u^2 - a^2}} = \cosh^{-1}(u/a) + C$ for u> a > 0
6.: $\displaystyle\int \frac{du}{a^2 - u^2} = \left\{ \begin{array}{ll} (1/a) \tanh^... ...< a^2 \\ (1/a) \coth^{-1}(u/a) + C & \mbox{ if } u^2 > a^2 \end{array} \right.$
7.: $\displaystyle\int \frac{du}{u \sqrt{a^2 - u^2}} = -(1/a) \, \mbox{sech}^{-1}(u/a) + C$ for 0 < u < a
8.: $\displaystyle\int \frac{du}{u \sqrt{a^2 + u^2}} = -(1/a) \, \mbox{csch}^{-1} \vert u/a\vert + C$ for $u \neq 0$