Final answer:
To complete the square for the integral of x² - 20x + 84 dx, we rewrite the expression as (x - 10)² - 16 and then integrate to find \(\frac{1}{3}(x - 10)^3 - 16x + C\), where C is the constant of integration.
Step-by-step explanation:
To complete the square and find the indefinite integral of the function x² - 20x + 84 dx, we start by completing the square for the quadratic expression. This involves rearranging the quadratic into the form (x-h)² + k, where h and k are constants. The coefficient of x² is already 1, which is convenient. We then look at the coefficient of x, which is -20, and use it to find h by taking half of it and squaring the result. Therefore, h = -20 / 2 = -10, and when we square it, we get 100. Now we add and subtract 100 inside the quadratic to complete the square:
x² - 20x + 100 - 100 + 84 = (x - 10)² - 16
The integral we need to solve is now:
∫ (x - 10)² - 16 dx
We can integrate this by treating it as the sum of two separate integrals:
∫ (x - 10)² dx - ∫ 16 dx
The first integral is that of a squared term, and the second is a constant. Integrating these gives us:
\(\frac{1}{3}(x - 10)^3 - 16x + C\)
This is our final answer, where C represents the constant of integration.