What is the integral of 1/sqrt(1 - x^2) dx?

The integral of ( \frac{1}{\sqrt{1 - x^2}} , dx ) is a standard integral in calculus, often encountered in trigonometric contexts. The correct result of this integral is ( \arcsin(x) + C ).

This result can be understood by recognizing that the function ( \frac{1}{\sqrt{1 - x^2}} ) relates to the derivative of the arcsine function. Specifically, if you differentiate ( \arcsin(x) ), you obtain ( \frac{1}{\sqrt{1 - x^2}} ). Therefore, when you perform the integral of ( \frac{1}{\sqrt{1 - x^2}} ), you are essentially reversing this derivative process.

Additionally, the arcsine function is defined on the interval [-1, 1], where ( \sqrt{1 - x^2} ) is positive, ensuring that the integral is valid in this domain. This characteristic makes ( \arcsin(x) ) the appropriate choice as it directly corresponds to the antiderivative of the integrand.

In the context of the other options, ( \arccos(x) ) is related