Integral, from 0 to 1, of dx/root(1-x^2)

[leo255]

Homework Statement

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Integral, from 0 to 1, of dx/root(1-x^2)

Homework Equations

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d/dx of arcsin = 1/root(1-x^2)

The Attempt at a Solution

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?

 

[STEMucator]

Nothing is wrong.

 

[ehild]

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?

What is arcsin(1)?

 

[leo255]

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?

 

[Mark44]

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?

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:
$$\lim_{b \to 1^-}\int_0^b \frac{dx}{\sqrt{1 - x^2}}$$