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Would you consider it fair if ChatGPT answered using the WolframAlpha plugin? https://user-images.githubusercontent.com/13973198/231566582-6ee98a69-77c3-42ca-9

• hgsgm 4y ago
  Can you try it again without WA? I'm curious how often ChatGPT makes mistakes because its temperature makes it sometimes randomly give wrong answers intentionally, to be generatively creative.
  • JCharante 4y ago
    GPT3.5-turbo We can solve this integral by using integration by parts. Let u = arctan x and dv = x dx. Then we have: du/dx = 1/(1+x^2) v = (1/2)x^2 Using the formula for integration by parts, we get: ∫x arctan x dx = x (arctan x) - ∫(1/2)x^2 / (1+x^2) dx We can simplify the integral on the right-hand side by using the substitution u = 1+x^2, du/dx = 2x: ∫(1/2)x^2 / (1+x^2) dx = (1/2) ∫du/u = (1/2) ln|1+x^2| + C Putting it all together, we have: ∫x arctan x dx = x (arctan x) - (1/2) ln|1+x^2| + C where C is the constant of integration. GPT-4: To calculate the integral of x * arctan(x) dx, we'll use integration by parts, which states: ∫u dv = uv - ∫v du First, we need to choose our u and dv: u = arctan(x) dv = x dx Now, we differentiate u and integrate dv: du = (1 / (1 + x^2)) dx v = (1/2) x^2 Now, apply the integration by parts formula: ∫x arctan(x) dx = (1/2) x^2 arctan(x) - ∫(1/2) x^2 * (1 / (1 + x^2)) dx Let's simplify the second integral: ∫(1/2) x^2 * (1 / (1 + x^2)) dx = (1/2) ∫x^2 / (1 + x^2) dx Now, let's use a substitution method to solve this integral: Let z = x^2 + 1, so dz = 2x dx. Then, x^2 = z - 1 and x dx = (1/2) dz. We can rewrite the integral in terms of z: (1/2) ∫(z - 1) / z dz = (1/2) ∫(z / z - 1 / z) dz = (1/2) ∫(1 - 1/z) dz Now, integrate with respect to z: (1/2) [z - ln(|z|)] + C = (1/2) [(x^2 + 1) - ln(|x^2 + 1|)] + C Now, plug this back into our integration by parts expression: ∫x arctan(x) dx = (1/2) x^2 arctan(x) - (1/2) [(x^2 + 1) - ln(|x^2 + 1|)] + C So the final result is: (1/2) x^2 arctan(x) - (1/2) x^2 - (1/2) ln(|x^2 + 1|) + C
    • hgsgm 4y ago
      Can I ask for one more? In the prompt, tell it to check its work by differentiating it's answer, and try again if the derivative doesn't match.
      • JCharante 4y ago
        Prompt: Check your work by differentiating your answer GPT3.5: Sure! Let's differentiate our answer and see if we get back to the original integrand. ∫x arctan x dx = x (arctan x) - (1/2) ln|1+x^2| + C Taking the derivative with respect to x, we get: d/dx [x (arctan x) - (1/2) ln|1+x^2| + C] = arctan x + x (1/(1+x^2)) - (1/2) (2x/(1+x^2)) = arctan x + x/(1+x^2) - x/(1+x^2) = arctan x Therefore, the derivative of our answer is indeed x arctan x. This confirms that our answer is correct. GPT 4: Let's differentiate our result with respect to x: f(x) = (1/2) x^2 arctan(x) - (1/2) x^2 - (1/2) ln(|x^2 + 1|) + C f'(x) = d/dx [(1/2) x^2 arctan(x) - (1/2) x^2 - (1/2) ln(|x^2 + 1|) + C] We'll differentiate each term separately: d/dx [(1/2) x^2 arctan(x)] We use the product rule (fg)' = f'g + fg': f = (1/2) x^2, g = arctan(x) f' = x, g' = 1 / (1 + x^2) d/dx [(1/2) x^2 arctan(x)] = x * arctan(x) + (1/2) x^2 * (1 / (1 + x^2)) d/dx [-(1/2) x^2] (1/2) * 2x = -x d/dx [-(1/2) ln(|x^2 + 1|)] Using the chain rule: (1/2) * (1 / (x^2 + 1)) * (2x) = -x / (x^2 + 1) The derivative of the constant C is 0. Now, combine the derivatives of all terms: f'(x) = x * arctan(x) + (1/2) x^2 * (1 / (1 + x^2)) - x - x / (x^2 + 1) Now, notice that the second term and the last term can be combined: (1/2) x^2 * (1 / (1 + x^2)) - x / (x^2 + 1) = x(x^2 / 2) / (x^2 + 1) - x / (x^2 + 1) = x(x^2 - 2) / (2(x^2 + 1)) Now our derivative looks like: f'(x) = x * arctan(x) + x(x^2 - 2) / (2(x^2 + 1)) - x Factor out x: f'(x) = x[arctan(x) + (x^2 - 2) / (2(x^2 + 1)) - 1] The derivative of our answer is x * arctan(x), which matches the original integrand.
• sverona 4y ago
  Would you let your students use Mathematica on such an exam?
  • skeaker 4y ago
    If it was built into their consciousness, then yeah