To integrate $\frac{1}{1 + \sin x}$ with respect to $x$ using Simpson's rule with six strips over the interval $[0, \frac{\pi}{2}]$, follow these steps:

1. Determine the width of each strip: The interval is $[0, \frac{\pi}{2}]$ and we are using six strips. Thus, the width $h$ of each strip is: $h = \frac{\frac{\pi}{2} - 0}{6} = \frac{\pi}{12}$

2. Calculate the $x$-values: The $x$-values at the endpoints of each strip are: $x_0 = 0, \quad x_1 = \frac{\pi}{12}, \quad x_2 = \frac{\pi}{6}, \quad x_3 = \frac{\pi}{4}, \quad x_4 = \frac{5\pi}{12}, \quad x_5 = \frac{\pi}{2}, \quad x_6 = \frac{\pi}{2}$

3. Evaluate the function at these points: Calculate the function values at these points: $f(x) = \frac{1}{1 + \sin x}$ $f(x_0) = \frac{1}{1 + \sin 0} = 1$ $f(x_1) = \frac{1}{1 + \sin \left(\frac{\pi}{12}\right)}$ $f(x_2) = \frac{1}{1 + \sin \left(\frac{\pi}{6}\right)} = \frac{1}{1 + \frac{1}{2}} = \frac{2}{3}$ $f(x_3) = \frac{1}{1 + \sin \left(\frac{\pi}{4}\right)} = \frac{1}{1 + \frac{\sqrt{2}}{2}} = \frac{2}{2 + \sqrt{2}}$ $f(x_4) = \frac{1}{1 + \sin \left(\frac{5\pi}{12}\right)}$ $f(x_5) = \frac{1}{1 + \sin \left(\frac{\pi}{2}\right)} = \frac{1}{2}$ $f(x_6) = \frac{1}{1 + \sin \left(\frac{\pi}{2}\right)} = \frac{1}{2}$

4. Apply Simpson's rule: Simpson's rule is given by: $\int_{a}^{b} f(x) \, dx \approx \frac{h}{3} \left[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + 2f(x_4) + 4f(x_5) + f(x_6)\right]$

   Plugging in the values: $\int_{0}^{\frac{\pi}{2}} \frac{1}{1 + \sin x} \, dx \approx \frac{\frac{\pi}{12}}{3} \left[1 + 4f(x_1) + 2 \cdot \frac{2}{3} + 4 \cdot \frac{2}{2 + \sqrt{2}} + 2f(x_4) + 4 \cdot \frac{1}{2} + \frac{1}{2}\right]$

   Simplifying the coefficients: $= \frac{\pi}{36} \left[1 + 4f(x_1) + \frac{4}{3} + \frac{8}{2 + \sqrt{2}} + 2f(x_4) + 2 + \frac{1}{2}\right]$

5. Complete the evaluation: Substitute $f(x_1)$ and $f(x_4)$ with their values (or numerically compute): $f(x_1) \approx 0.927 \quad \text{and} \quad f(x_4) \approx 0.737$

   Hence: $= \frac{\pi}{36} \left[1 + 4 \cdot 0.927 + \frac{4}{3} + \frac{8}{2 + \sqrt{2}} + 2 \cdot 0.737 + 2 + \frac{1}{2}\right]$

   $= \frac{\pi}{36} \left[1 + 3.708 + 1.333 + \frac{8}{2 + \sqrt{2}} + 1.474 + 2 + 0.5\right]$

   $= \frac{\pi}{36} \left[1 + 3.708 + 1.333 + 1.860 + 1.474 + 2 + 0.5\right]$

   $= \frac{\pi}{36} \left[11.875\right]$

   $= \frac{11.875 \pi}{36}$

Therefore, the approximate value of the integral using Simpson's rule with six strips is: $\int_{0}^{\frac{\pi}{2}} \frac{1}{1 + \sin x} \, dx \approx 1.036$

Would you like more details or have any questions? Here are 5 other questions you might find interesting:

1. How to find the Taylor series expansion of a function?
2. What is the divergence theorem and how is it used in vector calculus?
3. How to solve a differential equation using the method of undetermined coefficients?
4. Can you explain the concept of eigenvalues and eigenvectors?
5. How to perform polynomial long division?

Tip: When using numerical methods like Simpson's rule, ensure that the function values are accurately computed to maintain precision in the final result.