User:JoelP

Integrals Involving $$t = \sqrt{a^2-x^2}$$

 * $$\int t \;dx = \frac{1}{2}\left(xt+a^2\sin^{-1}\frac{x}{a}\right) \qquad\mbox{(}|x|\leq|a|\mbox{)}$$


 * $$\int\frac{dx}{t} = \sin^{-1}\frac{x}{a} \qquad\mbox{(}|x|\leq|a|\mbox{)}$$


 * $$\int xt\;dx = -\frac{1}{3} t^3 \qquad\mbox{(}|x|\leq|a|\mbox{)}$$


 * $$\int\frac{t\;dx}{x} = t-a\ln\left|\frac{a+t}{x}\right| \qquad\mbox{(}|x|\leq|a|\mbox{)}$$


 * $$\int\frac{x^2\;dx}{t} = -\frac{x}{2}t+\frac{a^2}{2}\sin^{-1}\frac{x}{a} \qquad\mbox{(}|x|\leq|a|\mbox{)}$$


 * $$\int t\;dx = \frac{1}{2}\left(xt-\sgn x\,\cosh^{-1}\left|\frac{x}{a}\right|\right) \qquad\mbox{(for }|x|\ge|a|\mbox{)}$$