The discussion revolves around evaluating the integral from 0 to 1 of dx/sqrt(1-x^2), which relates to the arcsine function. Participants are exploring the implications of the Fundamental Theorem of Calculus in the context of an improper integral due to the behavior of the integrand at the upper limit.
Some participants confirm the need to take a limit as the upper bound approaches 1 from the left, indicating a productive direction in addressing the problem's complexities. There is acknowledgment of the integrand's behavior at the boundary, but no consensus on a complete resolution has been reached.
The discussion highlights the challenge posed by the integrand being undefined at x = 1, which raises questions about the application of standard integration techniques and the need for limits in this context.
leo255 said:Since d/dx of arcsin = 1/root(1-x^2), we have that the integral, from 0 to 1, of dx/root(1-x^2) equals to arcsin, from 0 to 1.
arcsin(1) - arcsin(0) = arcsin(1). I know I'm missing something here. What did I do wrong?
Yes, this should be done. The integrand is undefined at x = 1, so the Fund. Thm. of Calculus doesn't apply. You can get around this by evaluating this limit:leo255 said:arcsin(1) is pi/2. I asked someone in my class about this, and he said that I should be taking the limit, as b (or whatever other variable) approaches 1, from the left-hand side. Can you guys confirm if this is something that should be done for this problem?