Evaluate the definite integral: ${\int }_0^1\frac{2}{4}{t}^3-\frac{3}{4}{t}^2+\frac{2}{5}tdt$ from $t=0$ to $t=1$.

• A. $\frac{3}{10}$
• B. $\frac{1}{4}$
• C. $0$
• D. $\frac{1}{10}$

Answer: (D)

Verified step by step guidance

1. Step 1: Begin by identifying the integral to evaluate: \( \int_{0}^{1} \left( \frac{2}{4}t^3 - \frac{3}{4}t^2 + \frac{2}{5}t \right) \, dt \). Simplify the coefficients where possible, such as \( \frac{2}{4} = \frac{1}{2} \). The integral becomes \( \int_{0}^{1} \left( \frac{1}{2}t^3 - \frac{3}{4}t^2 + \frac{2}{5}t \right) \, dt \).
2. Step 2: Apply the power rule for integration to each term. Recall that the power rule states \( \int t^n \, dt = \frac{t^{n+1}}{n+1} \). For each term: \( \frac{1}{2}t^3 \) becomes \( \frac{1}{2} \cdot \frac{t^4}{4} \), \( -\frac{3}{4}t^2 \) becomes \( -\frac{3}{4} \cdot \frac{t^3}{3} \), and \( \frac{2}{5}t \) becomes \( \frac{2}{5} \cdot \frac{t^2}{2} \).
3. Step 3: Combine the results of the integration into a single expression: \( \frac{1}{8}t^4 - \frac{1}{4}t^3 + \frac{1}{5}t^2 \). This represents the antiderivative of the given function.
4. Step 4: Evaluate the definite integral by substituting the limits of integration \( t = 0 \) and \( t = 1 \) into the antiderivative. First, substitute \( t = 1 \): \( \frac{1}{8}(1)^4 - \frac{1}{4}(1)^3 + \frac{1}{5}(1)^2 \). Then, substitute \( t = 0 \): \( \frac{1}{8}(0)^4 - \frac{1}{4}(0)^3 + \frac{1}{5}(0)^2 \).
5. Step 5: Subtract the value of the antiderivative at \( t = 0 \) from the value at \( t = 1 \). This will yield the final result of the definite integral. Simplify the terms to find the numerical value.