Which of the following limits is equal to the definite integral from $2$ to $5$ of $1+x^4$ $dx$?

• A. $\underset{n\to \infty }{lim}\sum i=1n[1+(2+\frac{3i}{n}\left){\left(4\right)}^{}\right]\cdot \frac{3}{n}$
• B. $\underset{n\to \infty }{lim}\sum i=1n[1+(2+\frac{i}{n}\left){\left(4\right)}^{}\right]\cdot \frac{1}{n}$
• C. $\underset{n\to \infty }{lim}\sum i=1n[1+(2+\frac{5i}{n}\left){\left(4\right)}^{}\right]\cdot \frac{5}{n}$
• D. $\underset{n\to \infty }{lim}\sum i=1n[1+(5+\frac{3i}{n}\left){\left(4\right)}^{}\right]\cdot \frac{3}{n}$

Answer: (A)

Verified step by step guidance

1. Step 1: Recall the definition of a definite integral as the limit of a Riemann sum. The definite integral from a to b of a function f(x) dx can be expressed as lim_{n \to \infty} \sum_{i=1}^n f(x_i) \cdot \Delta x, where \Delta x = \frac{b-a}{n} and x_i represents the sample points within each subinterval.
2. Step 2: Identify the interval of integration. In this problem, the definite integral is from 2 to 5, so the interval is [2, 5]. The width of each subinterval is \Delta x = \frac{5-2}{n} = \frac{3}{n}.
3. Step 3: Determine the sample points x_i within the interval. A common choice for sample points is the left endpoint of each subinterval, which can be expressed as x_i = 2 + i \cdot \Delta x = 2 + \frac{3i}{n}.
4. Step 4: Substitute the function f(x) = 1 + x^4 into the Riemann sum formula. The sum becomes \sum_{i=1}^n \left[1 + \left(2 + \frac{3i}{n}\right)^4\right] \cdot \Delta x, where \Delta x = \frac{3}{n}.
5. Step 5: Compare the given options to the derived Riemann sum expression. The correct limit is lim_{n \to \infty} \sum_{i=1}^n \left[1 + \left(2 + \frac{3i}{n}\right)^4\right] \cdot \frac{3}{n}, as it matches the interval, sample points, and function provided in the problem.