$\int{1\over \sqrt{1-x^2}}\,dx$
How do I solve this without $\sin$ or $\cos$ substitution? I want to try using $u$ substitution because I feel like it will be more intuitive for me. So far, I have tried to make the function to the power of a negative exponent:
${(\sqrt{1-x^2})}^{-1}$
I tried to do $u$ $=$ $1-x^2$, and do the substitution method, but I keep getting the wrong answer; I'm trying to use this indefinite integral to solve a definite one from $1/2$ to $\sqrt{3}$/$2$.
I'd rather not use trigonometric substitution because I want to do this assuming I don't already know that this is the derivative of $\sin^{-1}x$.