On dividing the interval into 10 equal parts and applying Simpson's 1/3 rule, find the value of the integral from 0 to 5 of (5 / (4x + 5)) dx correct to 4 decimal places.

Steps to Solve

1. Identify the integral and the limits

   The integral we need to evaluate is

   $$ I = \int_0^5 \frac{5}{4x + 5} , dx $$

   We will divide the interval ([0, 5]) into (n = 10) equal parts.

2. Calculate the width of each subinterval

   The width (h) of each subinterval is calculated as:

   $$ h = \frac{b - a}{n} = \frac{5 - 0}{10} = 0.5 $$

3. Determine the (x) values for the subintervals

   The (x) values are given by:

   $$ x_i = a + ih \text{ for } i = 0, 1, 2, \ldots, n $$

   So we get:

   • (x_0 = 0)
   • (x_1 = 0.5)
   • (x_2 = 1.0)
   • (x_3 = 1.5)
   • (x_4 = 2.0)
   • (x_5 = 2.5)
   • (x_6 = 3.0)
   • (x_7 = 3.5)
   • (x_8 = 4.0)
   • (x_9 = 4.5)
   • (x_{10} = 5.0)

4. Calculate the function values at each (x)

   Evaluate the function (f(x) = \frac{5}{4x + 5}) at each (x_i):

   • (f(x_0) = f(0) = 1)
   • (f(x_1) = f(0.5) = \frac{5}{7})
   • (f(x_2) = f(1.0) = 1)
   • (f(x_3) = f(1.5) = \frac{10}{14} = \frac{5}{7})
   • (f(x_4) = f(2.0) = 1)
   • (f(x_5) = f(2.5) = \frac{5}{15} = \frac{1}{3})
   • (f(x_6) = f(3.0) = \frac{5}{20} = \frac{1}{4})
   • (f(x_7) = f(3.5) = \frac{5}{24})
   • (f(x_8) = f(4.0) = \frac{1}{5})
   • (f(x_9) = f(4.5) = \frac{5}{32})
   • (f(x_{10}) = f(5.0) = \frac{1}{4})

5. Apply Simpson's 1/3 Rule

   Using Simpson's 1/3 Rule, we have:

   $$ I \approx \frac{h}{3} \left[ f(x_0) + 4(f(x_1) + f(x_3) + f(x_5) + f(x_7) + f(x_9)) + 2(f(x_2) + f(x_4) + f(x_6) + f(x_8) + f(x_{10})) \right] $$

   Substitute in the values calculated above:

   $$ I \approx \frac{0.5}{3} \left[ 1 + 4\left( \frac{5}{7} + \frac{10}{14} + \frac{1}{3} + \frac{5}{24} + \frac{5}{32} \right) + 2\left( 1 + 1 + \frac{1}{4} + \frac{1}{5} + \frac{1}{4} \right) \right] $$

   Carefully calculate this expression.

6. Evaluate the numerical expression

   After calculating the numerical expression step-by-step, we find:

   • Calculate (f(x_1), f(x_3), f(x_5), f(x_7), f(x_9))
   • Calculate (f(x_2), f(x_4), f(x_6), f(x_8), f(x_{10}))
   • Sum everything and multiply by (\frac{h}{3})

7. Round the result

   Round the final result to four decimal places.