To solve this problem, you will need to understand the principles of variable changes in integration, as well as how to calculate definite and antiderivative integrals.
Let's start with the first part of the problem- finding the definite integral of the function sqrt(1 - x²) from 0 to 1.
Step 1: Identify the function to integrate.
The function given is sqrt(1 - x²), which we can write as (1 - x²)^(1/2).
Step 2: Calculate the definite integral.
To calculate the definite integral, we integrate the function (1 - x²)^(1/2) with respect to 'x' from 0 to 1. Upon integration, we find that the definite integral is π/4.
Now, let's move on to the second part of the problem - finding the antiderivative of the expression (x - 4)(x² - 8).
Step 1: Identify the expression to integrate.
The expression given is (x - 4)(x² - 8), which can be expanded to x^3 - 4x² - 8x + 32.
Step 2: Calculate the antiderivative.
To calculate the antiderivative, we integrate the function x^3 - 4x² - 8x + 32 with respect to 'x'. Upon integration, we find that the antiderivative is x^4/4 - 4x³/3 - 4x² + 32x.
So, the results are: the definite integral of sqrt(1 - x²) from 0 to 1 is π/4, and the antiderivative of the expression (x - 4)(x² - 8) is x^4/4 - 4x^3/3 - 4x² + 32x.
It should be noted that every step in the process of integration requires careful application of the formulas and rules of calculus. Specifically, the Fundamental Theorem of Calculus, the Power Rule, and the rules for integrating polynomials are used in this problem. Make sure you are comfortable with these principles before attempting problems like this one.