Understanding the Derivative of Inverse Trig Functions

[alijan kk]

Homework Statement

why this formula works ?

Homework Equations

The Attempt at a Solution

when i take the derivative of the right side ,,, there is an additional "a" in the numerator in place of 1,, why the derivative of arcsine of (u/a) not exactly same with the expression under the integral sign

 

[andrewkirk]

In order to finally perform the integration in the last step you need to do a substitution, say replacing $u/a$ by $x$ so that your integrand has the form $1/\sqrt{1-x^2}$. In order to do that you need to make the other changes that are part of doing a variable substitution in integration. Those changes should cancel out the factor $1/a$ and leave you with the nice, clean result you want.

 

[alijan kk]

now i understand it very well,,,, thankyou for they reply

In order to finally perform the integration in the last step you need to do a substitution, say replacing $u/a$ by $x$ so that your integrand has the form $1/\sqrt{1-x^2}$. In order to do that you need to make the other changes that are part of doing a variable substitution in integration. Those changes should cancel out the factor $1/a$ and leave you with the nice, clean result