Evaluate : `int _1^(3) (x^(2) +5x) dx ` , expressing as a limit of sum.

Answer

To evaluate the integral \( \int_1^3 (x^2 + 5x) \, dx \) and express it as a limit of sums, we can follow these steps:

### Step 1: Set Up the Integral
We start with the integral:
\[ I = \int_1^3 (x^2 + 5x) \, dx \]

### Step 2: Find the Antiderivative
The antiderivative of \( x^2 \) is \( \frac{x^3}{3} \) and the antiderivative of \( 5x \) is \( \frac{5x^2}{2} \). Therefore, the antiderivative of \( x^2 + 5x \) is:
\[ F(x) = \frac{x^3}{3} + \frac{5x^2}{2} \]

### Step 3: Evaluate the Antiderivative at the Limits
Now we evaluate \( F(x) \) at the upper limit (3) and lower limit (1):
\[ F(3) = \frac{3^3}{3} + \frac{5 \cdot 3^2}{2} = \frac{27}{3} + \frac{5 \cdot 9}{2} = 9 + \frac{45}{2} = 9 + 22.5 = 31.5 \]
\[ F(1) = \frac{1^3}{3} + \frac{5 \cdot 1^2}{2} = \frac{1}{3} + \frac{5}{2} = \frac{1}{3} + 2.5 = \frac{1}{3} + \frac{7.5}{3} = \frac{8.5}{3} \]

### Step 4: Calculate the Definite Integral
Now we can calculate the definite integral:
\[ I = F(3) - F(1) = 31.5 - \frac{8.5}{3} \]
To perform this subtraction, we convert \( 31.5 \) to a fraction:
\[ 31.5 = \frac{63}{2} \]
Now we need a common denominator to subtract:
\[ \frac{63}{2} - \frac{8.5}{3} = \frac{63 \cdot 3}{6} - \frac{8.5 \cdot 2}{6} = \frac{189}{6} - \frac{17}{6} = \frac{172}{6} = \frac{86}{3} \]

### Final Result
Thus, the value of the integral is:
\[ I = \frac{86}{3} \]

### Expressing as a Limit of Sums
To express the integral as a limit of sums, we can use the Riemann sum approach. We divide the interval \([1, 3]\) into \( n \) equal parts, each of width \( \Delta x = \frac{3 - 1}{n} = \frac{2}{n} \). The points in the interval can be expressed as:
\[ x_i = 1 + i \Delta x = 1 + i \cdot \frac{2}{n} \]
The Riemann sum \( S_n \) can be expressed as:
\[ S_n = \sum_{i=1}^{n} f(x_i) \Delta x = \sum_{i=1}^{n} \left( \left(1 + \frac{2i}{n}\right)^2 + 5\left(1 + \frac{2i}{n}\right) \right) \cdot \frac{2}{n} \]
Taking the limit as \( n \to \infty \):
\[ I = \lim_{n \to \infty} S_n \]