Evaluate the definite integral: ${\int }_1^4(t^3-t^2-4) dt$.

• A. $28$
• B. $36$
• C. $48$
• D. $36.75$

Answer: (B)

Verified step by step guidance

1. Step 1: Recall the formula for evaluating a definite integral: \( \int_{a}^{b} f(t) \, dt = F(b) - F(a) \), where \( F(t) \) is the antiderivative of \( f(t) \).
2. Step 2: Find the antiderivative of the integrand \( t^3 - t^2 - 4 \). Use the power rule for integration: \( \int t^n \, dt = \frac{t^{n+1}}{n+1} \) for \( n \neq -1 \). The antiderivative is \( \frac{t^4}{4} - \frac{t^3}{3} - 4t \).
3. Step 3: Apply the limits of integration \( t = 1 \) and \( t = 4 \) to the antiderivative. Substitute \( t = 4 \) into \( F(t) = \frac{t^4}{4} - \frac{t^3}{3} - 4t \) to compute \( F(4) \).
4. Step 4: Substitute \( t = 1 \) into \( F(t) = \frac{t^4}{4} - \frac{t^3}{3} - 4t \) to compute \( F(1) \).
5. Step 5: Subtract \( F(1) \) from \( F(4) \) to find the value of the definite integral: \( \int_{1}^{4} (t^3 - t^2 - 4) \, dt = F(4) - F(1) \).