Question

What is the result of ∫(1/√(1 - x^2))dx?

A) Arcsin(x) + C
B) Arccos(x) + C
C) −Arcsin(x) + k
D) −Arccos(x) + k

Answer 1

The integral ∫(1/√(1 - x^2))dx is solved using the antiderivative of the integrand, resulting in Arcsin(x) + C as the answer, which corresponds to Option A.

Step-by-step explanation:

The student asked about the result of the integral ∫(1/√(1 - x^2))dx. This integral is a standard result in calculus that corresponds to the inverse trigonometric functions. Specifically, this integral is equal to the arcsine function, or arcsin(x), as it is one of the antiderivatives of the integrand 1/√(1 - x^2). Therefore, the answer to the integral is Arcsin(x) + C, where C represents the constant of integration.

To evaluate this integral, we recognize that the derivative of arcsin(x) is 1/√(1 - x^2). Thus, the indefinite integral or antiderivative of 1/√(1 - x^2) with respect to x is arcsin(x) plus a constant. This antiderivative is crucial in understanding and solving problems involving trigonometric functions and their inverses within calculus.

The final answer to the student's question is Option A) Arcsin(x) + C.