integrate1/1+sinx with respect to x with upper limit pi/2 and lower limit 0 using Simpson's rule with six strips

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?
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4. Can you explain the concept of eigenvalues and eigenvectors?
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**Tip:** When using numerical methods like Simpson's rule, ensure that the function values are accurately computed to maintain precision in the final result.

Learn how to integrate the function 1/(1 + sin(x)) with respect to x from 0 to pi/2 using Simpson's rule with six strips. Follow a step-by-step solution and numerical computation to approximate the integral value. Suitable for advanced college-level students studying numerical integration methods.

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