| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To approximate the integral $\int_{0}^4 \cos(1 + \sqrt{x}) dx$ using Simpson's rule with $n=8$, we first divide the interval $[0, 4]$ into $n=8$ subintervals, each of length $\Delta x = \frac{4-0}{8} = \frac{1}{2}$. The endpoints of the subintervals are $x_0=0, x_1=\frac{1}{2}, x_2=1, x_3=\frac{3}{2}, x_4=2, x_5=\frac{5}{2}, x_6=3, x_7=\frac{7}{2}, x_8=4$. Next, we evaluate the function $f(x) = \cos(1 + \sqrt{x})$ at these points: $f(x_0) = \cos(1 + \sqrt{0}) = \cos(1) \approx 0.5403$ $f(x_1) = \cos(1 + \sqrt{\frac{1}{2}}) \approx 0.2705$ $f(x_2) = \cos(1 + \sqrt{1}) = \cos(2) \approx -0.4161$ $f(x_3) = \cos(1 + \sqrt{\frac{3}{2}}) \approx -0.6536$ $f(x_4) = \cos(1 + \sqrt{2}) \approx -0.7510$ $f(x_5) = \cos(1 + \sqrt{\frac{5}{2}}) \approx -0.6536$ $f(x_6) = \cos(1 + \sqrt{3}) \approx -0.4161$ $f(x_7) = \cos(1 + \sqrt{\frac{7}{2}}) \approx 0.2705$ $f(x_8) = \cos(1 + \sqrt{4}) = \cos(3) \approx 0.9894$ Finally, we apply Simpson's rule: $\int_{0}^4 \cos(1 + \sqrt{x}) dx \approx \frac{\Delta x}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + 2f(x_4) + 4f(x_5) + 2f(x_6) + 4f(x_7) + f(x_8)]$ $\approx \frac{1}{6} [0.5403 + 4(0.2705) + 2(-0.4161) + 4(-0.6536) + 2(-0.7510) + 4(-0.6536) + 2(-0.4161) + 4(0.2705) + 0.9894]$ $\approx \frac{1}{6} [-2.6144] \approx -0.4357$ Therefore, the answer is $\boxed{-0.4357}$. |
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| deepseekr1 | To approximate the value of the integral \(\int_{0}^4 \cos(1 + \sqrt{x}) \, dx\) using Simpson's rule with \( n = 8 \): 1. **Determine the interval and step size**: - The interval is \([0, 4]\). - The number of subintervals \( n = 8 \), so the step size \( \Delta x = \frac{4 - 0}{8} = 0.5 \). 2. **List the points and compute function values**: - The points are \( x_0 = 0, x_1 = 0.5, x_2 = 1.0, x_3 = 1.5, x_4 = 2.0, x_5 = 2.5, x_6 = 3.0, x_7 = 3.5, x_8 = 4.0 \). - Compute \( f(x_i) = \cos(1 + \sqrt{x_i}) \) for each \( x_i \). 3. **Apply Simpson's rule coefficients**: - The coefficients for \( n = 8 \) are \( 1, 4, 2, 4, 2, 4, 2, 4, 1 \). 4. **Calculate the function values and multiply by coefficients**: - \( f(0) = \cos(1) \approx 0.540302 \) - \( f(0.5) = \cos(1 + \sqrt{0.5}) \approx -0.123290 \) multiplied by 4: \( -0.49316 \) - \( f(1.0) = \cos(2) \approx -0.416147 \) multiplied by 2: \( -0.832294 \) - \( f(1.5) = \cos(1 + \sqrt{1.5}) \approx -0.623496 \) multiplied by 4: \( -2.493984 \) - \( f(2.0) = \cos(1 + \sqrt{2}) \approx -0.767543 \) multiplied by 2: \( -1.535086 \) - \( f(2.5) = \cos(1 + \sqrt{2.5}) \approx -0.854272 \) multiplied by 4: \( -3.417088 \) - \( f(3.0) = \cos(1 + \sqrt{3}) \approx -0.916716 \) multiplied by 2: \( -1.833432 \) - \( f(3.5) = \cos(1 + \sqrt{3.5}) \approx -0.958032 \) multiplied by 4: \( -3.832128 \) - \( f(4.0) = \cos(3) \approx -0.989992 \) 5. **Sum the values and multiply by \(\Delta x / 3\)**: - Sum: \( 0.540302 - 0.49316 - 0.832294 - 2.493984 - 1.535086 - 3.417088 - 1.833432 - 3.832128 - 0.989992 \approx -14.886833 \) - Multiply by \( \frac{0.5}{3} \approx 0.16666667 \): \( -14.886833 \times 0.16666667 \approx -2.481143598 \) The approximate value of the integral using Simpson's rule with \( n = 8 \) is \(\boxed{-2.4811}\). |
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| gpto1 | To approximate the value of the integral \[ \int_{0}^{4} \cos(1 + \sqrt{x})\,dx \] using Simpson's Rule with \( n = 8 \), we will follow these steps: **Step 1: Determine \( h \) and the partition points** Simpson's Rule requires dividing the interval \([a, b]\) into \( n \) equal subintervals, where \( n \) is even. Given: - \( a = 0 \) - \( b = 4 \) - \( n = 8 \) Compute \( h \): \[ h = \frac{b - a}{n} = \frac{4 - 0}{8} = 0.5 \] Compute the partition points \( x_i \): \[ x_i = a + i h, \quad i = 0,1,2,\dots,8 \] So: \[ x_0 = 0,\ x_1 = 0.5,\ x_2 = 1.0,\ x_3 = 1.5,\ x_4 = 2.0,\ x_5 = 2.5,\ x_6 = 3.0,\ x_7 = 3.5,\ x_8 = 4.0 \] **Step 2: Compute \( f(x_i) = \cos(1 + \sqrt{x_i}) \)** Create a table to compute \( \sqrt{x_i} \), \( s_i = 1 + \sqrt{x_i} \), and \( f(x_i) = \cos(s_i) \): | \( i \) | \( x_i \) | \( \sqrt{x_i} \) | \( s_i = 1 + \sqrt{x_i} \) | \( f(x_i) = \cos(s_i) \) | |---------|-----------|------------------|---------------------------|----------------------------| | 0 | 0.0 | 0.0 | 1.0 | 0.54030231 | | 1 | 0.5 | 0.70710678 | 1.70710678 | \(-0.134523\) | | 2 | 1.0 | 1.0 | 2.0 | \(-0.41614684\) | | 3 | 1.5 | 1.22474487 | 2.22474487 | \(-0.615375\) | | 4 | 2.0 | 1.41421356 | 2.41421356 | \(-0.748613\) | | 5 | 2.5 | 1.58113883 | 2.58113883 | \(-0.846957\) | | 6 | 3.0 | 1.73205081 | 2.73205081 | \(-0.915314\) | | 7 | 3.5 | 1.87082869 | 2.87082869 | \(-0.957854\) | | 8 | 4.0 | 2.0 | 3.0 | \(-0.9899925\) | **Step 3: Apply Simpson's Rule** Simpson's Rule formula for \( n = 8 \) is: \[ \int_{a}^{b} f(x)\,dx \approx \frac{h}{3} \left[ f(x_0) + 4\sum_{\text{odd }i} f(x_i) + 2\sum_{\text{even }i,\ i\ne 0,n} f(x_i) + f(x_n) \right] \] Compute the sums: - Sum of \( f(x_0) \) and \( f(x_8) \): \[ S_1 = f(x_0) + f(x_8) = 0.54030231 + (-0.9899925) = -0.44969019 \] - Sum of \( 4 \times f(x_i) \) for odd \( i \): \[ S_2 = 4 \times \left[ f(x_1) + f(x_3) + f(x_5) + f(x_7) \right] = 4 \times \left[ -0.134523 - 0.615375 - 0.846957 - 0.957854 \right] = 4 \times (-2.554709) = -10.218836 \] - Sum of \( 2 \times f(x_i) \) for even \( i \), excluding \( x_0 \) and \( x_8 \): \[ S_3 = 2 \times \left[ f(x_2) + f(x_4) + f(x_6) \right] = 2 \times \left[ -0.41614684 - 0.748613 - 0.915314 \right] = 2 \times (-2.08007384) = -4.16014768 \] Compute the total sum: \[ S = S_1 + S_2 + S_3 = -0.44969019 - 10.218836 - 4.16014768 = -14.82867387 \] **Step 4: Calculate the approximate value** Finally, compute the approximate value of the integral: \[ \int_{0}^{4} \cos(1 + \sqrt{x})\,dx \approx \frac{h}{3} \times S = \frac{0.5}{3} \times (-14.82867387) = \frac{1}{6} \times (-14.82867387) = -2.471445645 \] **Answer:** Using Simpson's Rule with \( n = 8 \), the approximate value of the integral is: \[ \int_{0}^{4} \cos(1 + \sqrt{x})\,dx \approx -2.4714 \] |
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| GPT-4o | To approximate the value of the integral \(\int_{0}^4 \cos(1 + \sqrt{x}) \, dx\) using Simpson's rule with \(n = 8\), we need to follow these steps: 1. **Determine the step size \(h\):** Since \(n = 8\) and the interval is from 0 to 4, the step size \(h\) is calculated as: \[ h = \frac{b - a}{n} = \frac{4 - 0}{8} = \frac{4}{8} = 0.5 \] 2. **Calculate the function values at the endpoints and intermediate points:** We need to evaluate the function \(f(x) = \cos(1 + \sqrt{x})\) at the points \(x_i = a + i \cdot h\) for \(i = 0, 1, 2, \dots, 8\). \[ \begin{aligned} x_0 &= 0, & f(x_0) &= \cos(1 + \sqrt{0}) = \cos(1) \\ x_1 &= 0.5, & f(x_1) &= \cos(1 + \sqrt{0.5}) \\ x_2 &= 1, & f(x_2) &= \cos(1 + \sqrt{1}) = \cos(2) \\ x_3 &= 1.5, & f(x_3) &= \cos(1 + \sqrt{1.5}) \\ x_4 &= 2, & f(x_4) &= \cos(1 + \sqrt{2}) \\ x_5 &= 2.5, & f(x_5) &= \cos(1 + \sqrt{2.5}) \\ x_6 &= 3, & f(x_6) &= \cos(1 + \sqrt{3}) \\ x_7 &= 3.5, & f(x_7) &= \cos(1 + \sqrt{3.5}) \\ x_8 &= 4, & f(x_8) &= \cos(1 + \sqrt{4}) = \cos(3) \end{aligned} \] 3. **Apply Simpson's rule:** Simpson's rule for \(n\) subintervals is given by: \[ \int_{a}^{b} f(x) \, dx \approx \frac{h}{3} \left[ f(x_0) + 4 \sum_{i \text{ odd}} f(x_i) + 2 \sum_{i \text{ even}, i \ne 0, n} f(x_i) + f(x_n) \right] \] Here, \(h = 0.5\), and \(x_i\) values are already calculated. Substituting these into Simpson's rule formula: \[ \int_{0}^{4} \cos(1 + \sqrt{x}) \, dx \approx \frac{0.5}{3} \left[ f(x_0) + 4(f(x_1) + f(x_3) + f(x_5) + f(x_7)) + 2(f(x_2) + f(x_4) + f(x_6)) + f(x_8) \right] \] Let's list the function values: \[ \begin{aligned} f(x_0) &= \cos(1) \\ f(x_1) &= \cos(1 + \sqrt{0.5}) \\ f(x_2) &= \cos(2) \\ f(x_3) &= \cos(1 + \sqrt{1.5}) \\ f(x_4) &= \cos(1 + \sqrt{2}) \\ f(x_5) &= \cos(1 + \sqrt{2.5}) \\ f(x_6) &= \cos(1 + \sqrt{3}) \\ f(x_7) &= \cos(1 + \sqrt{3.5}) \\ f(x_8) &= \cos(3) \end{aligned} \] Now, plug these values into the formula: \[ \int_{0}^{4} \cos(1 + \sqrt{x}) \, dx \approx \frac{0.5}{3} \left[ \cos(1) + 4(\cos(1 + \sqrt{0.5}) + \cos(1 + \sqrt{1.5}) + \cos(1 + \sqrt{2.5}) |
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| llama3.1-70B-Instruct | To approximate the value of the integral $\int_{0}^4 cos(1 + \sqrt{x}) dx$ using Simpson's rule with $n=8$, we first need to understand Simpson's rule and how to apply it. Simpson's rule is a method for approximating the value of a definite integral. It states that for a function $f(x)$ and an even number of subintervals $n$, the integral from $a$ to $b$ can be approximated as: $$\int_{a}^{b} f(x) dx \approx \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n) \right]$$ where $h = \frac{b-a}{n}$ is the width of each subinterval, and $x_i = a + ih$ for $i = 0, 1, 2, \ldots, n$. Given $n=8$, we have $h = \frac{4-0}{8} = \frac{1}{2}$. The points $x_i$ are therefore $0, \frac{1}{2}, 1, \frac{3}{2}, 2, \frac{5}{2}, 3, \frac{7}{2}, 4$. Now, we need to calculate $f(x_i) = cos(1 + \sqrt{x_i})$ for each $x_i$: 1. $f(0) = cos(1 + \sqrt{0}) = cos(1)$ 2. $f(\frac{1}{2}) = cos(1 + \sqrt{\frac{1}{2}})$ 3. $f(1) = cos(1 + \sqrt{1}) = cos(2)$ 4. $f(\frac{3}{2}) = cos(1 + \sqrt{\frac{3}{2}})$ 5. $f(2) = cos(1 + \sqrt{2})$ 6. $f(\frac{5}{2}) = cos(1 + \sqrt{\frac{5}{2}})$ 7. $f(3) = cos(1 + \sqrt{3})$ 8. $f(\frac{7}{2}) = cos(1 + \sqrt{\frac{7}{2}})$ 9. $f(4) = cos(1 + \sqrt{4}) = cos(3)$ Let's calculate these values: 1. $cos(1) \approx 0.540302$ 2. $cos(1 + \sqrt{\frac{1}{2}}) \approx 0.509525$ 3. $cos(2) \approx -0.416147$ 4. $cos(1 + \sqrt{\frac{3}{2}}) \approx 0.426511$ 5. $cos(1 + \sqrt{2}) \approx 0.309017$ 6. $cos(1 + \sqrt{\frac{5}{2}}) \approx 0.198912$ 7. $cos(1 + \sqrt{3}) \approx 0.087266$ 8. $cos(1 + \sqrt{\frac{7}{2}}) \approx -0.020785$ 9. $cos(3) \approx -0.989992$ Now, we apply Simpson's rule: $$\int_{0}^4 cos(1 + \sqrt{x}) dx \approx \frac{1/2}{3} \left[ 0.540302 + 4(0.509525) + 2(-0.416147) + 4(0.426511) + 2(0.309017) + 4(0.198912) + 2(0.087266) + 4(-0.020785) + (-0.989992) \right]$$ $$\approx \frac{1}{6} \left[ 0.540302 + 2.0381 - 0.832294 + 1.706044 + 0.618034 + 0.795648 + 0.174532 - 0.08314 - 0.989992 \right]$$ $$\approx \frac{1}{6} \left[ 3.926424 \right]$$ $$\approx 0.090240404$$ Therefore, the approximation of the integral $\int_{0}^4 cos(1 + \sqrt{x}) dx$ using Simpson's rule with $n=8$ is approximately $0.090240404$. |
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| llama3.1-405B-Instruct | To approximate the value of $\int_{0}^4 \cos(1 + \sqrt{x}) dx$ using Simpson's rule with $n=8$, we first need to divide the interval $[0,4]$ into $8$ equal subintervals. The width of each subinterval is $\frac{4-0}{8} = 0.5$. Next, we need to calculate the function values at the points $x_0=0$, $x_1=0.5$, $x_2=1$, $x_3=1.5$, $x_4=2$, $x_5=2.5$, $x_6=3$, $x_7=3.5$, and $x_8=4$. Using a calculator or computer, we can calculate the function values: $$\begin{array}{c|c} x & \cos(1 + \sqrt{x}) \\ \hline 0 & 0.540302 \\ 0.5 & 0.577582 \\ 1 & 0.604524 \\ 1.5 & 0.621117 \\ 2 & 0.630957 \\ 2.5 & 0.635567 \\ 3 & 0.637423 \\ 3.5 & 0.638035 \\ 4 & 0.638285 \end{array}$$ Now, we can apply Simpson's rule: $$\int_{0}^4 \cos(1 + \sqrt{x}) dx \approx \frac{0.5}{3} \left[ 0.540302 + 4(0.577582 + 0.621117 + 0.635567) + 2(0.604524 + 0.630957 + 0.637423) + 0.638285 \right]$$ Simplifying, we get: $$\int_{0}^4 \cos(1 + \sqrt{x}) dx \approx 2.42671$$ Therefore, the approximate value of the integral using Simpson's rule with $n=8$ is $2.42671$. |
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| DeepSeek-Coder-V2 | To approximate the value of the integral \(\int_{0}^4 \cos(1 + \sqrt{x}) \, dx\) using Simpson's rule with \(n = 8\), we follow these steps: 1. **Determine the interval width \(h\):** \[ h = \frac{b - a}{n} = \frac{4 - 0}{8} = 0.5 \] 2. **Define the function \(f(x) = \cos(1 + \sqrt{x})\).** 3. **Calculate the values of the function at the points \(x_i = a + ih\) for \(i = 0, 1, 2, \ldots, 8\):** \[ \begin{aligned} &x_0 = 0, \quad f(x_0) = \cos(1 + \sqrt{0}) = \cos(1) \\ &x_1 = 0.5, \quad f(x_1) = \cos(1 + \sqrt{0.5}) \\ &x_2 = 1, \quad f(x_2) = \cos(1 + \sqrt{1}) = \cos(2) \\ &x_3 = 1.5, \quad f(x_3) = \cos(1 + \sqrt{1.5}) \\ &x_4 = 2, \quad f(x_4) = \cos(1 + \sqrt{2}) \\ &x_5 = 2.5, \quad f(x_5) = \cos(1 + \sqrt{2.5}) \\ &x_6 = 3, \quad f(x_6) = \cos(1 + \sqrt{3}) \\ &x_7 = 3.5, \quad f(x_7) = \cos(1 + \sqrt{3.5}) \\ &x_8 = 4, \quad f(x_8) = \cos(1 + \sqrt{4}) = \cos(3) \\ \end{aligned} \] 4. **Apply Simpson's rule:** \[ \int_{0}^{4} \cos(1 + \sqrt{x}) \, dx \approx \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + 2f(x_4) + 4f(x_5) + 2f(x_6) + 4f(x_7) + f(x_8) \right] \] 5. **Substitute the values:** \[ \begin{aligned} &\int_{0}^{4} \cos(1 + \sqrt{x}) \, dx \approx \frac{0.5}{3} \left[ \cos(1) + 4\cos(1 + \sqrt{0.5}) + 2\cos(2) + 4\cos(1 + \sqrt{1.5}) + 2\cos(1 + \sqrt{2}) + 4\cos(1 + \sqrt{2.5}) + 2\cos(1 + \sqrt{3}) + 4\cos(1 + \sqrt{3.5}) + \cos(3) \right] \end{aligned} \] 6. **Calculate the numerical values:** \[ \begin{aligned} &\cos(1) \approx 0.540302 \\ &\cos(1 + \sqrt{0.5}) \approx 0.497571 \\ &\cos(2) \approx -0.416147 \\ &\cos(1 + \sqrt{1.5}) \approx 0.110345 \\ &\cos(1 + \sqrt{2}) \approx -0.070737 \\ &\cos(1 + \sqrt{2.5}) \approx -0.324631 \\ &\cos(1 + \sqrt{3}) \approx -0.514148 \\ &\cos(1 + \sqrt{3.5}) \approx -0.631267 \\ &\cos(3) \approx -0.989992 \\ \end{aligned} \] 7. **Sum the values and multiply by \(\frac{0.5}{3}\):** \[ \begin{aligned} &\text{Sum} = 0.540302 + 4 \times 0.497571 + 2 \times -0.416147 + 4 \times 0.110345 + 2 \times -0.070737 + 4 \times -0.324631 + 2 \times -0.514148 + 4 \times -0.631267 + -0.989992 \\ &\text{Sum} \approx 0.540302 + 1.990284 - 0.832294 + 0.44138 + -0.141474 - 1.298524 - 1.028296 - 2.525068 - 0.989992 \\ &\text{Sum} \approx -4.843674 \\ &\int_{0}^{4} \cos(1 + \sqrt{x}) \, dx \approx \frac{0.5}{3} \times -4.843674 \approx -0.807279 \end{aligned} \] Thus, the approximate value of the integral using Simpson's rule with \(n = 8\) is: \[ \boxed{-0.807279} \] |
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| Qwen2-72B-Instruct | To approximate the integral $\int_{0}^4 \cos(1 + \sqrt{x}) dx$ using Simpson's rule with $n=8$, we first divide the interval $[0, 4]$ into $n=8$ subintervals, each of length $\Delta x = \frac{4-0}{8} = \frac{1}{2}$. The endpoints of the subintervals are $x_0=0, x_1=\frac{1}{2}, x_2=1, x_3=\frac{3}{2}, x_4=2, x_5=\frac{5}{2}, x_6=3, x_7=\frac{7}{2}, x_8=4$. Next, we evaluate the function $f(x) = \cos(1 + \sqrt{x})$ at these points: $f(x_0) = \cos(1 + \sqrt{0}) = \cos(1) \approx 0.5403$ $f(x_1) = \cos(1 + \sqrt{\frac{1}{2}}) \approx 0.2705$ $f(x_2) = \cos(1 + \sqrt{1}) = \cos(2) \approx -0.4161$ $f(x_3) = \cos(1 + \sqrt{\frac{3}{2}}) \approx -0.6536$ $f(x_4) = \cos(1 + \sqrt{2}) \approx -0.7510$ $f(x_5) = \cos(1 + \sqrt{\frac{5}{2}}) \approx -0.6536$ $f(x_6) = \cos(1 + \sqrt{3}) \approx -0.4161$ $f(x_7) = \cos(1 + \sqrt{\frac{7}{2}}) \approx 0.2705$ $f(x_8) = \cos(1 + \sqrt{4}) = \cos(3) \approx 0.9894$ Finally, we apply Simpson's rule: $\int_{0}^4 \cos(1 + \sqrt{x}) dx \approx \frac{\Delta x}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + 2f(x_4) + 4f(x_5) + 2f(x_6) + 4f(x_7) + f(x_8)]$ $\approx \frac{1}{6} [0.5403 + 4(0.2705) + 2(-0.4161) + 4(-0.6536) + 2(-0.7510) + 4(-0.6536) + 2(-0.4161) + 4(0.2705) + 0.9894]$ $\approx \frac{1}{6} [-2.6144] \approx -0.4357$ Therefore, the answer is $\boxed{-0.4357}$. |
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| baichuan4 | To approximate the value of the integral $\int_{0}^4 \cos(1 + \sqrt{x}) dx$ using Simpson's rule with $n=8$, we first need to divide the interval $[0, 4]$ into $n$ subintervals of equal length. Since $n=8$, each subinterval has length $\frac{4-0}{8} = 0.5$. Next, we evaluate the function $\cos(1 + \sqrt{x})$ at the endpoints of these subintervals, which will be the points $x_0, x_1, \ldots, x_8$, where $x_i = 0 + 0.5i$ for $i=0,1,\ldots,8$. The Simpson's rule formula for approximating the integral is: $$ \int_{a}^{b} f(x) dx \approx \frac{\Delta x}{3} \left[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \ldots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n)\right] $$ where $\Delta x = \frac{b-a}{n}$. Now, let's calculate the function values at the specified points: - $f(x_0) = f(0) = \cos(1 + \sqrt{0}) = \cos(1)$ - $f(x_1) = f(0.5) = \cos(1 + \sqrt{0.5})$ - $f(x_2) = f(1) = \cos(1 + \sqrt{1})$ - $f(x_3) = f(1.5) = \cos(1 + \sqrt{1.5})$ - $f(x_4) = f(2) = \cos(1 + \sqrt{2})$ - $f(x_5) = f(2.5) = \cos(1 + \sqrt{2.5})$ - $f(x_6) = f(3) = \cos(1 + \sqrt{3})$ - $f(x_7) = f(3.5) = \cos(1 + \sqrt{3.5})$ - $f(x_8) = f(4) = \cos(1 + \sqrt{4}) = \cos(3)$ Plugging these values into the Simpson's rule formula with $\Delta x = 0.5$, we get: $$ \int_{0}^4 \cos(1 + \sqrt{x}) dx \approx \frac{0.5}{3} \left[\cos(1) + 4\cos(1 + \sqrt{0.5}) + 2\cos(1 + \sqrt{1}) + 4\cos(1 + \sqrt{1.5}) + \ldots + 2\cos(1 + \sqrt{3}) + 4\cos(1 + \sqrt{3.5}) + \cos(3)\right] $$ Now, we just need to add up all these cosine values with the appropriate coefficients and multiply by $\frac{0.5}{3}$ to get the final approximation. Let's calculate: - $\cos(1) \approx 0.540302$ - $\cos(1 + \sqrt{0.5}) \approx 0.371787$ - $\cos(1 + \sqrt{1}) \approx -0.416147$ - $\cos(1 + \sqrt{1.5}) \approx -0.836001$ - $\cos(1 + \sqrt{2}) \approx -0.989992$ - $\cos(1 + \sqrt{2.5}) \approx -0.968583$ - $\cos(1 + \sqrt{3}) \approx -0.707107$ - $\cos(1 + \sqrt{3.5}) \approx -0.270646$ - $\cos(3) \approx -0.989992$ Putting it all together: $$ \int_{0}^4 \cos(1 + \sqrt{x}) dx \approx \frac{0.5}{3} [0.540302 + 4(0.371787) + 2(-0.416147) + 4(-0.836001) + 2(-0.989992) + 4(-0.968583) + 2(-0.707107) + 4(-0.270646) -0.989992] $$ $$ \int_{0}^4 \cos(1 + \sqrt{x}) dx \approx \frac{0.5}{3} [0.540302 + 1.487148 - 0.832294 - 3.344004 - 1.979984 - 3.874332 - 1.414214 - 1.082584 - 0.989992] $$ $$ \int_{0}^4 \cos(1 + \sqrt{x}) dx \approx \frac{0.5}{3} [-11.519744] $$ $$ \int_{0}^4 \cos(1 + \sqrt{x}) dx \approx -1.919957 $$ So the approximated value of the integral using Simpson's rule with $n=8$ is $-1.919957$. |
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| doubao | The Simpson's rule formula is: $\int_{a}^{b}f(x)dx \approx \frac{b-a}{3n}[f(a) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(b)]$ Here, $a=0$, $b=4$, $n=8$. The step size $h = \frac{b-a}{n} = \frac{4}{8} = 0.5$. The $x$ values are $0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4$. Evaluating the function at these points: $f(0) = \cos(1+\sqrt{0}) = \cos(1)$; $f(0.5) = \cos(1+\sqrt{0.5})$; $f(1) = \cos(1+\sqrt{1}) = \cos(2)$; $f(1.5) = \cos(1+\sqrt{1.5})$; $f(2) = \cos(1+\sqrt{2})$; $f(2.5) = \cos(1+\sqrt{2.5})$; $f(3) = \cos(1+\sqrt{3})$; $f(3.5) = \cos(1+\sqrt{3.5})$; $f(4) = \cos(1+\sqrt{4}) = \cos(3)$. Using the Simpson's rule: $\int_{0}^4 \cos(1 + \sqrt{x}) dx \approx \frac{4-0}{3\times8}[\cos(1) + 4\cos(1+\sqrt{0.5}) + 2\cos(2) + 4\cos(1+\sqrt{1.5}) + 2\cos(2.5) + 4\cos(1+\sqrt{2}) + 2\cos(3) + 4\cos(1+\sqrt{3}) + \cos(3.5) + \cos(3)]$. Evaluating this expression will give the approximation. |
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| minimax | To approximate the value of the integral $\int_{0}^4 \cos(1 + \sqrt{x}) dx$ using Simpson's rule, we first need to divide the interval [0, 4] into n equal subintervals. Since we are given n=8, we will divide the interval into 8 subintervals, each of width $\Delta x = \frac{4 - 0}{8} = 0.5$. The Simpson's rule formula for approximating the integral of a function f(x) over an interval [a, b] using n subintervals is given by: $$ \int_{a}^{b} f(x) dx \approx \frac{\Delta x}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \ldots + 4f(x_{n-1}) + f(x_n) \right] $$ where $x_0, x_1, x_2, \ldots, x_n$ are the endpoints of the subintervals, with $x_0 = a$ and $x_n = b$. For our case, the subintervals are [0, 0.5], [0.5, 1], [1, 1.5], ..., [3.5, 4]. Let's calculate the values of the function at these points: $$ f(x) = \cos(1 + \sqrt{x}) $$ $$ f(0) = \cos(1 + \sqrt{0}) = \cos(1) $$ $$ f(0.5) = \cos(1 + \sqrt{0.5}) $$ $$ f(1) = \cos(1 + \sqrt{1}) = \cos(2) $$ $$ f(1.5) = \cos(1 + \sqrt{1.5}) $$ $$ f(2) = \cos(1 + \sqrt{2}) $$ $$ f(2.5) = \cos(1 + \sqrt{2.5}) $$ $$ f(3) = \cos(1 + \sqrt{3}) $$ $$ f(3.5) = \cos(1 + \sqrt{3.5}) $$ $$ f(4) = \cos(1 + \sqrt{4}) = \cos(3) $$ Now, we can apply Simpson's rule: $$ \int_{0}^{4} \cos(1 + \sqrt{x}) dx \approx \frac{0.5}{3} \left[ \cos(1) + 4\cos(1 + \sqrt{0.5}) + 2\cos(1 + \sqrt{1}) + 4\cos(1 + \sqrt{1.5}) + \ldots + 4\cos(1 + \sqrt{3.5}) + \cos(3) \right] $$ We can now calculate the values of the cosine function at the specified points and sum them up according to the pattern of Simpson's rule: $$ \approx \frac{0.5}{3} \left[ \cos(1) + 4\cos(1.7071) + 2\cos(2) + 4\cos(2.4142) + 2\cos(2.7321) + 4\cos(3) + 2\cos(3.1623) + 4\cos(3.3541) + \cos(3) \right] $$ Now, you would typically use a calculator or a computer to evaluate the cosine function at these points and perform the arithmetic to get the final approximation. Since I cannot compute these values, I'll leave the expression as is. You can then substitute the numerical values of the cosines to get the final result. |
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| kimi | Simpson's rule is a method for numerical integration that approximates the definite integral of a function. It is based on the idea of approximating the function by a quadratic polynomial over each subinterval of the partition of the interval of integration. Simpson's rule states that for a function f(x) and an interval [a, b] divided into n equal subintervals, the integral can be approximated as: $$\int_{a}^{b} f(x) dx \approx \frac{\Delta x}{3} \left[ f(a) + 4f\left(a + \frac{\Delta x}{2}\right) + 2f(a + \Delta x) + 4f\left(a + \frac{3\Delta x}{2}\right) + \ldots + 2f(b - \Delta x) + f(b) \right]$$ where $\Delta x = \frac{b - a}{n}$. For the given problem, we want to approximate the integral $\int_{0}^4 \cos(1 + \sqrt{x}) dx$ using Simpson's rule with n=8. First, we need to calculate $\Delta x$: $$\Delta x = \frac{4 - 0}{8} = \frac{1}{2}$$ Now, we will evaluate the function at the necessary points: - $f(0) = \cos(1 + \sqrt{0}) = \cos(1)$ - $f\left(\frac{1}{2}\right) = \cos(1 + \sqrt{\frac{1}{2}})$ - $f(1) = \cos(1 + \sqrt{1}) = \cos(2)$ - $f\left(\frac{3}{2}\right) = \cos(1 + \sqrt{\frac{3}{2}})$ - $f(2) = \cos(1 + \sqrt{2})$ - $f\left(\frac{5}{2}\right) = \cos(1 + \sqrt{\frac{5}{2}})$ - $f(3) = \cos(1 + \sqrt{3})$ - $f\left(\frac{7}{2}\right) = \cos(1 + \sqrt{\frac{7}{2}})$ - $f(4) = \cos(1 + \sqrt{4}) = \cos(3)$ Now, we apply Simpson's rule: $$\int_{0}^4 \cos(1 + \sqrt{x}) dx \approx \frac{\frac{1}{2}}{3} \left[ \cos(1) + 4\cos(1 + \sqrt{\frac{1}{2}}) + 2\cos(2) + 4\cos(1 + \sqrt{\frac{3}{2}}) + 2\cos(1 + \sqrt{2}) + 4\cos(1 + \sqrt{\frac{5}{2}}) + 2\cos(3) + \cos(3) \right]$$ Let's calculate the values: $$\int_{0}^4 \cos(1 + \sqrt{x}) dx \approx \frac{1}{6} \left[ \cos(1) + 4\cos(1 + \sqrt{\frac{1}{2}}) + 2\cos(2) + 4\cos(1 + \sqrt{\frac{3}{2}}) + 2\cos(1 + \sqrt{2}) + 4\cos(1 + \sqrt{\frac{5}{2}}) + 2\cos(3) + \cos(3) \right]$$ Now, we can plug in the values of the cosine function at these points and calculate the approximation: $$\int_{0}^4 \cos(1 + \sqrt{x}) dx \approx \frac{1}{6} \left[ \cos(1) + 4\cos(1 + \frac{\sqrt{2}}{2}) + 2\cos(2) + 4\cos(1 + \sqrt{\frac{3}{2}}) + 2\cos(1 + \sqrt{2}) + 4\cos(1 + \frac{\sqrt{10}}{2}) + 2\cos(3) + \cos(3) \right]$$ This is the final formula for the approximation. To get a numerical value, you would need to calculate the cosines using a calculator or a computer algebra system. |
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| wenxinyiyan | To approximate the value of the integral $\int_{0}^4 \cos(1 + \sqrt{x}) dx$ using Simpson's rule with $n=8$ intervals, we follow these steps: Step 1: Determine the interval width $h$. The interval $[0, 4]$ is divided into $n=8$ equal subintervals. Therefore, the width of each subinterval is: $$h = \frac{4 - 0}{8} = 0.5$$ Step 2: Apply Simpson's rule. Simpson's rule approximates the integral over a given interval by using quadratic polynomials to approximate the function. The general formula for Simpson's rule with $n$ subintervals is: $$\int_{a}^{b} f(x) dx \approx \frac{h}{3} \left[ f(a) + 4f\left(a + \frac{h}{2}\right) + 2f(a + h) + 4f\left(a + \frac{3h}{2}\right) + \ldots + 4f\left(a + \frac{(n-1)h}{2}\right) + f(b) \right]$$ In our case, $a=0$, $b=4$, $h=0.5$, and $n=8$. We apply the formula to our function $f(x) = \cos(1 + \sqrt{x})$: $$\int_{0}^{4} \cos(1 + \sqrt{x}) dx \approx \frac{0.5}{3} \left[ \cos(1 + \sqrt{0}) + 4\cos\left(1 + \sqrt{0.5}\right) + 2\cos(1 + \sqrt{1}) + 4\cos\left(1 + \sqrt{1.5}\right) + \ldots + 4\cos\left(1 + \sqrt{3.5}\right) + \cos(1 + \sqrt{4}) \right]$$ Calculating each term and summing them up with the appropriate weights, we get the approximation for the integral. Note: The actual calculation of each term and the final summation is not shown here due to its length and complexity. It is recommended to use a calculator or a programming language to evaluate the expression accurately. Step 3: Evaluate the expression. By evaluating the above expression, we obtain the approximate value of the integral. Please note that without a calculator or programming environment, it is difficult to provide an exact numerical answer. However, following the above steps with the help of a suitable tool should yield a reasonably accurate approximation of the integral. |
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| zhipu | To approximate the value of the integral \(\int_{0}^4 \cos(1 + \sqrt{x}) \, dx\) using Simpson's rule with \(n = 8\), follow these steps: ### Step 1: Define the function Let \( f(x) = \cos(1 + \sqrt{x}) \). ### Step 2: Determine the interval width Given \( n = 8 \) and the interval \([0, 4]\), the width \( h \) of each subinterval is calculated as: \[ h = \frac{b - a}{n} = \frac{4 - 0}{8} = 0.5 \] ### Step 3: Compute the x-values and corresponding function values The x-values at the endpoints and midpoints of the subintervals are: \[ x_0 = 0, \, x_1 = 0.5, \, x_2 = 1, \, x_3 = 1.5, \, x_4 = 2, \, x_5 = 2.5, \, x_6 = 3, \, x_7 = 3.5, \, x_8 = 4 \] Evaluate the function \( f(x) \) at these points: \[ \begin{align*} f(x_0) &= f(0) = \cos(1 + \sqrt{0}) = \cos(1) \\ f(x_1) &= f(0.5) = \cos(1 + \sqrt{0.5}) \\ f(x_2) &= f(1) = \cos(1 + \sqrt{1}) = \cos(2) \\ f(x_3) &= f(1.5) = \cos(1 + \sqrt{1.5}) \\ f(x_4) &= f(2) = \cos(1 + \sqrt{2}) \\ f(x_5) &= f(2.5) = \cos(1 + \sqrt{2.5}) \\ f(x_6) &= f(3) = \cos(1 + \sqrt{3}) \\ f(x_7) &= f(3.5) = \cos(1 + \sqrt{3.5}) \\ f(x_8) &= f(4) = \cos(1 + \sqrt{4}) = \cos(3) \end{align*} \] ### Step 4: Apply Simpson's rule Simpson's rule is given by: \[ \int_{a}^{b} f(x) \, dx \approx \frac{h}{3} \left[ f(x_0) + 4 \sum_{i=1,3,5,\ldots}^{n-1} f(x_i) + 2 \sum_{i=2,4,6,\ldots}^{n-2} f(x_i) + f(x_n) \right] \] For \( n = 8 \): \[ \int_{0}^{4} \cos(1 + \sqrt{x}) \, dx \approx \frac{0.5}{3} \left[ f(x_0) + 4(f(x_1) + f(x_3) + f(x_5) + f(x_7)) + 2(f(x_2) + f(x_4) + f(x_6)) + f(x_8) \right] \] ### Step 5: Substitute the function values \[ \begin{align*} \int_{0}^{4} \cos(1 + \sqrt{x}) \, dx &\approx \frac{0.5}{3} \left[ \cos(1) + 4(\cos(1 + \sqrt{0.5}) + \cos(1 + \sqrt{1.5}) + \cos(1 + \sqrt{2.5}) + \cos(1 + \sqrt{3.5})) \right. \\ &\quad \left. + 2(\cos(2) + \cos(1 + \sqrt{2}) + \cos(1 + \sqrt{3})) + \cos(3) \right] \end{align*} \] ### Step 6: Calculate the numerical values Using a calculator or computer to find the values of the cosines: \[ \begin{align*} \cos(1) &\approx 0.5403 \\ \cos(1 + \sqrt{0.5}) &\approx \cos(1.7071) \approx -0.1294 \\ \cos(2) &\approx -0.4161 \\ \cos(1 + \sqrt{1.5}) &\approx \cos(2.2247) \approx -0.6700 \\ \cos(1 + \sqrt{2}) &\approx \cos(2.4142) \approx -0.7431 \\ \cos(1 + \sqrt{2.5}) &\approx \cos(2.5811) \approx -0.8090 \\ \cos(1 + \sqrt{3}) &\approx \cos(2.7321) \approx -0.8602 \\ \cos(1 + \sqrt{3.5}) &\approx \cos(2.8704) \approx -0.9042 \\ \cos(3) &\approx -0.9899 \end{align*} \] ### Step 7: Sum the values \[ \begin{align*} \int_{0}^{4} \cos(1 + \sqrt{x}) \, dx &\approx \frac{0.5}{3} \left[ 0.5403 + 4(-0.1294 - 0.6700 - 0.8090 - 0.9042) + 2(-0.4161 - 0.7431 - 0.8602) - 0.9899 \right] \\ &\approx \frac{0.5}{3} \left[ 0.5403 + 4(-2.5136) + 2(-2.0194) - 0.9899 \right] \\ &\approx \frac{0.5}{3} \left[ 0.5403 - 10.0544 - 4.0388 - 0.9899 \right] \\ &\approx \frac{0.5}{3} \left[ -14.5538 \right] \\ &\approx \frac{0.5}{3} \times -14.5538 \\ &\approx -2.4256 \end{align*} \] Thus, the approximate value of the integral \(\int_{0}^4 \cos(1 + \sqrt{x}) \, dx\) using Simpson's rule with \(n = 8\) is \(-2.4256\). |
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| gpto1 | To approximate the value of the integral \[ \int_{0}^{4} \cos(1 + \sqrt{x})\,dx \] using Simpson's Rule with \( n = 8 \), we will follow these steps: **Step 1: Determine \( h \) and the partition points** Simpson's Rule requires dividing the interval \([a, b]\) into \( n \) equal subintervals, where \( n \) is even. Given: - \( a = 0 \) - \( b = 4 \) - \( n = 8 \) Compute \( h \): \[ h = \frac{b - a}{n} = \frac{4 - 0}{8} = 0.5 \] Compute the partition points \( x_i \): \[ x_i = a + i h, \quad i = 0,1,2,\dots,8 \] So: \[ x_0 = 0,\ x_1 = 0.5,\ x_2 = 1.0,\ x_3 = 1.5,\ x_4 = 2.0,\ x_5 = 2.5,\ x_6 = 3.0,\ x_7 = 3.5,\ x_8 = 4.0 \] **Step 2: Compute \( f(x_i) = \cos(1 + \sqrt{x_i}) \)** Create a table to compute \( \sqrt{x_i} \), \( s_i = 1 + \sqrt{x_i} \), and \( f(x_i) = \cos(s_i) \): | \( i \) | \( x_i \) | \( \sqrt{x_i} \) | \( s_i = 1 + \sqrt{x_i} \) | \( f(x_i) = \cos(s_i) \) | |---------|-----------|------------------|---------------------------|----------------------------| | 0 | 0.0 | 0.0 | 1.0 | 0.54030231 | | 1 | 0.5 | 0.70710678 | 1.70710678 | \(-0.134523\) | | 2 | 1.0 | 1.0 | 2.0 | \(-0.41614684\) | | 3 | 1.5 | 1.22474487 | 2.22474487 | \(-0.615375\) | | 4 | 2.0 | 1.41421356 | 2.41421356 | \(-0.748613\) | | 5 | 2.5 | 1.58113883 | 2.58113883 | \(-0.846957\) | | 6 | 3.0 | 1.73205081 | 2.73205081 | \(-0.915314\) | | 7 | 3.5 | 1.87082869 | 2.87082869 | \(-0.957854\) | | 8 | 4.0 | 2.0 | 3.0 | \(-0.9899925\) | **Step 3: Apply Simpson's Rule** Simpson's Rule formula for \( n = 8 \) is: \[ \int_{a}^{b} f(x)\,dx \approx \frac{h}{3} \left[ f(x_0) + 4\sum_{\text{odd }i} f(x_i) + 2\sum_{\text{even }i,\ i\ne 0,n} f(x_i) + f(x_n) \right] \] Compute the sums: - Sum of \( f(x_0) \) and \( f(x_8) \): \[ S_1 = f(x_0) + f(x_8) = 0.54030231 + (-0.9899925) = -0.44969019 \] - Sum of \( 4 \times f(x_i) \) for odd \( i \): \[ S_2 = 4 \times \left[ f(x_1) + f(x_3) + f(x_5) + f(x_7) \right] = 4 \times \left[ -0.134523 - 0.615375 - 0.846957 - 0.957854 \right] = 4 \times (-2.554709) = -10.218836 \] - Sum of \( 2 \times f(x_i) \) for even \( i \), excluding \( x_0 \) and \( x_8 \): \[ S_3 = 2 \times \left[ f(x_2) + f(x_4) + f(x_6) \right] = 2 \times \left[ -0.41614684 - 0.748613 - 0.915314 \right] = 2 \times (-2.08007384) = -4.16014768 \] Compute the total sum: \[ S = S_1 + S_2 + S_3 = -0.44969019 - 10.218836 - 4.16014768 = -14.82867387 \] **Step 4: Calculate the approximate value** Finally, compute the approximate value of the integral: \[ \int_{0}^{4} \cos(1 + \sqrt{x})\,dx \approx \frac{h}{3} \times S = \frac{0.5}{3} \times (-14.82867387) = \frac{1}{6} \times (-14.82867387) = -2.471445645 \] **Answer:** Using Simpson's Rule with \( n = 8 \), the approximate value of the integral is: \[ \int_{0}^{4} \cos(1 + \sqrt{x})\,dx \approx -2.4714 \] |
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| deepseekv3 | To approximate the integral \(\int_{0}^4 \cos(1 + \sqrt{x}) \, dx\) using Simpson's rule with \(n = 8\), follow these steps: ### Step 1: Determine the interval width \(h\) The interval \([0, 4]\) is divided into \(n = 8\) subintervals. The width of each subinterval is: \[ h = \frac{b - a}{n} = \frac{4 - 0}{8} = 0.5 \] ### Step 2: Compute the function values at the endpoints and midpoints The function to evaluate is \(f(x) = \cos(1 + \sqrt{x})\). Compute \(f(x)\) at the endpoints and midpoints of each subinterval: \[ x_0 = 0, \quad x_1 = 0.5, \quad x_2 = 1, \quad x_3 = 1.5, \quad x_4 = 2, \quad x_5 = 2.5, \quad x_6 = 3, \quad x_7 = 3.5, \quad x_8 = 4 \] \[ \begin{align*} f(x_0) &= \cos(1 + \sqrt{0}) = \cos(1) \\ f(x_1) &= \cos(1 + \sqrt{0.5}) \\ f(x_2) &= \cos(1 + \sqrt{1}) = \cos(2) \\ f(x_3) &= \cos(1 + \sqrt{1.5}) \\ f(x_4) &= \cos(1 + \sqrt{2}) \\ f(x_5) &= \cos(1 + \sqrt{2.5}) \\ f(x_6) &= \cos(1 + \sqrt{3}) \\ f(x_7) &= \cos(1 + \sqrt{3.5}) \\ f(x_8) &= \cos(1 + \sqrt{4}) = \cos(3) \\ \end{align*} \] ### Step 3: Apply Simpson's rule formula Simpson's rule formula for \(n = 8\) is: \[ \int_{a}^b f(x) \, dx \approx \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + 2f(x_4) + 4f(x_5) + 2f(x_6) + 4f(x_7) + f(x_8) \right] \] Substitute the function values and \(h = 0.5\): \[ \int_{0}^4 \cos(1 + \sqrt{x}) \, dx \approx \frac{0.5}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + 2f(x_4) + 4f(x_5) + 2f(x_6) + 4f(x_7) + f(x_8) \right] \] ### Step 4: Compute the numerical value Calculate the numerical value of the expression: \[ \int_{0}^4 \cos(1 + \sqrt{x}) \, dx \approx \frac{0.5}{3} \left[ \cos(1) + 4\cos(1 + \sqrt{0.5}) + 2\cos(2) + 4\cos(1 + \sqrt{1.5}) + 2\cos(1 + \sqrt{2}) + 4\cos(1 + \sqrt{2.5}) + 2\cos(1 + \sqrt{3}) + 4\cos(1 + \sqrt{3.5}) + \cos(3) \right] \] Using a calculator or software to evaluate the cosine terms, the approximate value of the integral is: \[ \int_{0}^4 \cos(1 + \sqrt{x}) \, dx \approx 1.086 \] ### Final Answer: \[ \boxed{1.086} \] |
-2.47160136 |