Step 1: Understanding the Concept:
To solve this definite integral, we use the method of substitution. Since the derivative of the term inside the square root ($x^2 + 4$) is proportional to the $x$ outside, substitution simplifies the expression into a basic power form.
Step 2: Key Formula or Approach:
1. Let $t = x^2 + 4$, then $dt = 2x \, dx$.
2. Change the limits: When $x=0$, $t=4$. When $x=1$, $t=5$.
3. Use $\int t^n \, dt = \frac{t^{n+1}}{n+1} + C$.
Step 3: Detailed Explanation:
Substitute $t = x^2 + 4 \implies dt = 2x \, dx \implies x \, dx = \frac{dt}{2}$.
The integral becomes:
\[ \int_{4}^{5} \sqrt{t} \cdot \frac{dt}{2} = \frac{1}{2} \int_{4}^{5} t^{1/2} \, dt \]
\[ = \frac{1}{2} \left[ \frac{t^{3/2}}{3/2} \right]_{4}^{5} = \frac{1}{2} \cdot \frac{2}{3} \left[ t^{3/2} \right]_{4}^{5} \]
\[ = \frac{1}{3} \left[ 5^{3/2} - 4^{3/2} \right] \]
\[ = \frac{1}{3} [5\sqrt{5} - (2^2)^{3/2}] = \frac{1}{3} [5\sqrt{5} - 2^3] = \frac{1}{3} [5\sqrt{5} - 8] \]
Step 4: Final Answer:
The value of the integral is \( \frac{1}{3}[5\sqrt{5} - 8] \).