Question

Which of the following is an antiderivative of f(x) = √(1 - x³)?

A) (2/3)(1 - x³)(³/²)
B) (2/5)(1 - x³)(⁵/²)
C) (3/4)(1 - x³)(⁴/³)
D) (1/2)(1 - x³)(¹/²)

Answer 1

The antiderivative of the function f(x) = √(1 - x³) is found using the power rule for integration. The correct antiderivative is (A) (2/3)(1 - x³)^(3/2).

Step-by-step explanation:

The student is asking to identify the antiderivative of the function f(x) = √(1 - x³). The correct answer should be obtained by integrating the given function. We will use the power rule for integration, which states that the integral of x^n dx is (x^(n+1))/(n+1) + C, where C is the constant of integration, and n ≠ -1.

Applying the power rule, we get:

1. Recognize the inner function is (1 - x³) which we will integrate with respect to x.
2. The integral of f(x) = √(1 - x³) dx is the same as integrating (1 - x³)^(1/2) dx.
3. Using the power rule, an antiderivative of (1 - x³)^(1/2) is (2/3)⋅(1 - x³)^(3/2).

Comparing our result with the options given in the question, the correct antiderivative is (A) (2/3)(1 - x³)^(3/2).