Evaluate the definite integral: ${\int }_0^1(6x^2-6x+8) dx$.

• A. $7$
• B. $6$
• C. $8$
• D. $9$

Answer: (A)

Verified step by step guidance

1. Step 1: Recall the formula for evaluating a definite integral: \( \int_a^b f(x) \, dx = F(b) - F(a) \), where \( F(x) \) is the antiderivative of \( f(x) \).
2. Step 2: Find the antiderivative of the integrand \( 6x^2 - 6x + 8 \). Use the power rule for integration: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} \), and integrate each term separately.
3. Step 3: Apply the antiderivative to each term: \( \int 6x^2 \, dx = 2x^3 \), \( \int -6x \, dx = -3x^2 \), and \( \int 8 \, dx = 8x \). Combine these results to get \( F(x) = 2x^3 - 3x^2 + 8x \).
4. Step 4: Evaluate \( F(x) \) at the upper limit \( x = 1 \) and the lower limit \( x = 0 \). Substitute \( x = 1 \) into \( F(x) \) to find \( F(1) \), and substitute \( x = 0 \) into \( F(x) \) to find \( F(0) \).
5. Step 5: Compute the definite integral by subtracting \( F(0) \) from \( F(1) \): \( \int_0^1 (6x^2 - 6x + 8) \, dx = F(1) - F(0) \).