A definite integral by parts: every AI model's reply, tested

Solve this math problem step by step. Number the steps, check the result, and end with a line that starts "Final answer:".

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Evaluate the integral of x·e^x from x = 0 to x = 1.

How the AI math solver's test inputs are scored, with every model's results on all five.

1. GPT-6 Luna

   Passed: Final answer 1: right.

   1. Use integration by parts with u = x and dv = e^x dx.
      Then du = dx and v = e^x.
   
   2. Apply the integration-by-parts formula.
      ∫ x·e^x dx = x·e^x − ∫ e^x dx
      = x·e^x − e^x
      = e^x(x − 1)
   
   3. Evaluate the antiderivative at the bounds.
      ∫_0^1 x·e^x dx = [e^x(x − 1)]_0^1
      = e^1(1 − 1) − e^0(0 − 1)
      = 0 − (−1)
      = 1
   
   4. Check by differentiating the antiderivative.
      d/dx [e^x(x − 1)] = e^x(x − 1) + e^x = x·e^x
      So the antiderivative is correct, and the evaluation gives 1.
   
   Final answer: 1

   346 tokens in, 255 out (25 of them reasoning) · 2.9 s · $0.0001 · 1 message on Pro · answered by openai/gpt-6-luna via OpenAI · September 28, 2026

2. GLM 5.3 Flash

   Passed: Final answer 1: right.

   Reading: compute the definite integral ∫ from 0 to 1 of x·e^x dx.
   
   Step 1: Use integration by parts with u = x and dv = e^x dx.
   Then du = dx and v = e^x.
   
   Step 2: Apply ∫ u dv = u·v − ∫ v du:
   ∫ x·e^x dx = x·e^x − ∫ e^x dx.
   
   Step 3: Integrate the remaining term:
   ∫ e^x dx = e^x, so the antiderivative is x·e^x − e^x = e^x(x − 1) + C.
   
   Step 4: Evaluate from 0 to 1:
   At x = 1: e^1(1 − 1) = e·0 = 0.
   At x = 0: e^0(0 − 1) = 1·(−1) = −1.
   
   Step 5: Subtract:
   0 − (−1) = 1.
   
   Check: Estimate numerically — x·e^x rises from 0 to e ≈ 2.718, so the average is plausibly around 1 over [0,1]; exact antiderivative check: d/dx [e^x(x−1)] = e^x(x−1) + e^x = x·e^x. Correct.
   
   Final answer: 1

   354 tokens in, 304 out · 1.5 s · $0.0002 · 1 message on Pro · answered by z-ai/glm-5.3-flash via Friendli · September 28, 2026

3. Claude Haiku 5.5

   Passed: Final answer 1: right.

   1. We want the definite integral I = ∫ from 0 to 1 of x·e^x dx.
   
   2. Integrate by parts with u = x and dv = e^x dx.
      Then du = dx and v = e^x.
   
   3. Apply the formula ∫u dv = uv − ∫v du:
      ∫ x·e^x dx = x·e^x − ∫ e^x dx = x·e^x − e^x + C = e^x(x − 1) + C.
   
   4. Evaluate the antiderivative F(x) = e^x(x − 1) at the limits.
      F(1) = e^1(1 − 1) = e·0 = 0.
      F(0) = e^0(0 − 1) = 1·(−1) = −1.
   
   5. Subtract, by the Fundamental Theorem of Calculus:
      I = F(1) − F(0) = 0 − (−1) = 1.
   
   6. Check by a different method: expand e^x as a power series, e^x = sum over n≥0 of x^n/n!.
      Then x·e^x = sum over n≥0 of x^(n+1)/n!.
      Integrate term by term from 0 to 1: sum over n≥0 of 1/((n+1)·n!) = sum over n≥0 of 1/(n+1)!.
      The sum is 1/1! + 1/2! + 1/3! + ... = (e − 1) ... wait, we must be careful: sum over n≥0 of 1/(n+1)! = sum over k≥1 of 1/k! = e − 1.
      This gives e − 1 ≈ 1.718, which disagrees with step 5, so I recheck the series step.
   
   7. Recheck: x·e^x = sum over n≥0 of x^(n+1)/n!. Integrating x^(n+1) from 0 to 1 gives 1/(n+2), not 1/(n+1). So the term is 1/((n+2)·n!) = (n+1)/(n+2)!.
      Write (n+1)/(n+2)! = (n+2−1)/(n+2)! = 1/(n+1)! − 1/(n+2)!.
      Summing over n≥0 telescopes: the sum of 1/(n+1)! is e − 1, and the sum of 1/(n+2)! is e − 2. The difference is (e − 1) − (e − 2) = 1.
      This matches step 5.
   
   8. Sanity estimate: the integrand x·e^x is between 0 and e on [0,1], and it is small near 0 and about 2.72 at x = 1. Its average is roughly 1 over an interval of length 1, which is plausible.
   
   Final answer: 1

   454 tokens in, 866 out · 4.0 s · $0.0005 · 1 message on Pro · answered by anthropic/claude-haiku-5.5 via Anthropic · October 7, 2026

4. DeepSeek V4.1 Flash

   Passed: Final answer 1: right.

   1. We want the definite integral from x = 0 to x = 1 of x·e^x.
   
   2. Use integration by parts:
      Let u = x, so du = dx.
      Let dv = e^x dx, so v = e^x.
   
   3. The integration by parts formula is:
      integral(u dv) = u v - integral(v du)
   
   4. Substitute:
      integral(x e^x dx) = x e^x - integral(e^x dx)
   
   5. Since integral(e^x dx) = e^x, we get:
      integral(x e^x dx) = x e^x - e^x = e^x(x - 1)
   
   6. Evaluate from 0 to 1:
      At x = 1: e^1(1 - 1) = e·0 = 0
      At x = 0: e^0(0 - 1) = 1·(-1) = -1
   
   7. Subtract:
      0 - (-1) = 1
   
   8. Check: x·e^x is positive on [0, 1], so the integral must be positive. The value 1 is positive and reasonable since the curve is below about 2.72 and the interval length is 1.
   
   Final answer: 1

   371 tokens in, 417 out (150 of them reasoning) · 1.5 s · $0.0005 · 1 message on Pro · answered by deepseek/deepseek-v4.1-flash via Together · October 8, 2026

5. Gemini 3.8 Flash

   Passed: Final answer 1: right.

   Step 1: Identify the integration technique.
   The integral to evaluate is the definite integral of x*e^x dx from x = 0 to x = 1.
   We use integration by parts, which uses the formula:
   integral of u dv = u*v - integral of v du.
   
   Step 2: Choose u and dv.
   Let u = x, which gives du = dx.
   Let dv = e^x dx, which gives v = e^x.
   
   Step 3: Apply the integration by parts formula to find the indefinite integral.
   integral of x*e^x dx = x*e^x - integral of e^x dx
   integral of x*e^x dx = x*e^x - e^x + C
   integral of x*e^x dx = (x - 1)*e^x + C
   
   Step 4: Evaluate the antiderivative at the upper limit x = 1.
   (1 - 1)*e^1 = 0*e^1 = 0
   
   Step 5: Evaluate the antiderivative at the lower limit x = 0.
   (0 - 1)*e^0 = (-1)*1 = -1
   
   Step 6: Subtract the lower limit value from the upper limit value.
   Result = 0 - (-1)
   Result = 1
   
   Check the result:
   Differentiate the antiderivative (x - 1)*e^x:
   d/dx [(x - 1)*e^x] = 1*e^x + (x - 1)*e^x = e^x + x*e^x - e^x = x*e^x.
   The derivative matches the integrand, confirming the antiderivative is correct.
   Evaluating at 1 gives (1 - 1)*e = 0, and at 0 gives (0 - 1)*1 = -1.
   0 - (-1) = 1. The result is verified.
   
   Final answer: 1

   355 tokens in, 428 out · 2.5 s · $0.0009 · 1 message on Pro · answered by google/gemini-3.8-flash via Google AI Studio · September 28, 2026

6. GLM 5.3

   Passed: Final answer 1: right.

   Solve ∫ from 0 to 1 of x·e^x dx using integration by parts.
   
   Step 1: Choose u = x and dv = e^x dx, so du = dx and v = e^x.
   
   Step 2: Apply the integration by parts formula ∫ u dv = uv − ∫ v du:
   ∫ x·e^x dx = x·e^x − ∫ e^x dx
   
   Step 3: Integrate the remaining term:
   ∫ x·e^x dx = x·e^x − e^x + C = e^x(x − 1) + C
   
   Step 4: Evaluate from 0 to 1:
   [e^x(x − 1)] at x = 1 is e^1(0) = 0
   [e^x(x − 1)] at x = 0 is e^0(−1) = −1
   Integral = 0 − (−1) = 1
   
   Step 5: Check by an estimate: on [0,1], x·e^x lies between x and x·e, so the integral lies between 1/2 and e/2 ≈ 1.359. Our answer, 1, is in that range. Also verify the antiderivative by differentiating:
   d/dx [e^x(x − 1)] = e^x(x − 1) + e^x = x·e^x ✓
   
   Final answer: 1

   354 tokens in, 317 out · 1.8 s · $0.0014 · 1 message on Pro · answered by z-ai/glm-5.3 via Friendli · September 28, 2026

7. GPT-6.1 Sol

   Passed: Final answer 1: right.

   1. Use integration by parts:
   ∫ u dv = uv − ∫ v du
   
   Choose:
   u = x
   dv = e^x dx
   du = dx
   v = e^x
   
   2. Find the antiderivative:
   ∫ x·e^x dx = x·e^x − ∫ e^x dx
   ∫ x·e^x dx = x·e^x − e^x + C
   ∫ x·e^x dx = (x − 1)e^x + C
   
   3. Evaluate at the bounds:
   ∫ from 0 to 1 of x·e^x dx = [(x − 1)e^x] from 0 to 1
   = (1 − 1)e^1 − (0 − 1)e^0
   = 0 − (−1)
   = 1
   
   4. Check by differentiating the antiderivative:
   d/dx [(x − 1)e^x] = e^x + (x − 1)e^x
   = x·e^x
   
   This matches the integrand, confirming the calculation.
   
   Final answer: 1

   346 tokens in, 246 out · 2.9 s · $0.0016 · 1 message on Pro · answered by openai/gpt-6.1-sol via OpenAI · September 29, 2026

8. GPT-6 Sol

   Passed: Final answer 1: right.

   1. Use integration by parts with u = x and dv = e^x dx.
      du = dx
      v = e^x
   
   2. Apply the integration-by-parts formula.
      ∫x·e^x dx = x·e^x − ∫e^x dx
      ∫x·e^x dx = (x − 1)e^x + C
   
   3. Evaluate at the bounds.
      ∫(from 0 to 1) x·e^x dx = [(x − 1)e^x] from 0 to 1
      = (1 − 1)e^1 − (0 − 1)e^0
      = 0 − (−1)
      = 1
   
   4. Check by differentiating the antiderivative.
      d/dx [(x − 1)e^x] = e^x + (x − 1)e^x = x·e^x
   
   Final answer: 1

   346 tokens in, 276 out (62 of them reasoning) · 3.9 s · $0.0017 · 1 message on Pro · answered by openai/gpt-6-sol via OpenAI · September 28, 2026

9. Claude Haiku 4.5

   Passed: Final answer 1: right.

   I need to evaluate the definite integral:
   
   ∫[0 to 1] x·e^x dx
   
   Step 1: Set up integration by parts.
   For ∫ x·e^x dx, I'll use integration by parts: ∫ u dv = uv - ∫ v du
   Let u = x, so du = dx
   Let dv = e^x dx, so v = e^x
   
   Step 2: Apply integration by parts.
   ∫ x·e^x dx = x·e^x - ∫ e^x dx
   = x·e^x - e^x
   = e^x(x - 1)
   
   Step 3: Evaluate the antiderivative at the bounds.
   ∫[0 to 1] x·e^x dx = [e^x(x - 1)] evaluated from 0 to 1
   
   Step 4: Substitute x = 1.
   e^1(1 - 1) = e·0 = 0
   
   Step 5: Substitute x = 0.
   e^0(0 - 1) = 1·(-1) = -1
   
   Step 6: Compute the definite integral.
   [e^x(x - 1)] from 0 to 1 = 0 - (-1) = 1
   
   Step 7: Check the result by verifying the antiderivative.
   Take d/dx[e^x(x - 1)]:
   = e^x(x - 1) + e^x·1
   = e^x(x - 1 + 1)
   = e^x·x ✓
   
   This matches the integrand, so the antiderivative is correct.
   
   Final answer: 1

   365 tokens in, 413 out · 3.8 s · $0.0024 · 1 message on Pro · answered by anthropic/claude-haiku-4.5 via Anthropic · September 28, 2026

10. DeepSeek V4 Pro

    Passed: Final answer 1: right.

    Step 1. Use integration by parts with u = x and dv = e^x dx. Then du = dx and v = e^x.
    
    Step 2. The integration by parts formula gives:
    integral of x e^x dx = x e^x - integral of e^x dx.
    
    Step 3. Integrate the remaining term:
    integral of e^x dx = e^x, so
    integral of x e^x dx = x e^x - e^x = e^x (x - 1).
    
    Step 4. Evaluate from x = 0 to x = 1:
    [e^x (x - 1)] from 0 to 1
    = e^1 (1 - 1) - e^0 (0 - 1)
    = e * 0 - 1 * (-1)
    = 0 + 1
    = 1.
    
    Step 5. Check by differentiating e^x (x - 1):
    d/dx [e^x (x - 1)] = e^x (x - 1) + e^x = x e^x, which matches the original integrand.
    
    Final answer: 1

    345 tokens in, 628 out (394 of them reasoning) · 5.3 s · $0.0026 · 1 message on Pro · answered by deepseek/deepseek-v4-pro-0813 via Wafer · October 8, 2026

11. Claude Sonnet 5.5

    Passed: Final answer 1: right.

    We want the definite integral of x·e^x from 0 to 1. I'll use integration by parts.
    
    Step 1: Recall the integration by parts formula.
    integral of u dv = u·v - integral of v du
    
    Step 2: Choose u and dv.
    u = x, so du = dx
    dv = e^x dx, so v = e^x
    
    Step 3: Apply the formula to the indefinite integral.
    integral of x·e^x dx = x·e^x - integral of e^x dx
    
    Step 4: Integrate the remaining term.
    integral of x·e^x dx = x·e^x - e^x + C
    = (x - 1)·e^x + C
    
    Step 5: Evaluate from 0 to 1.
    [(x - 1)·e^x] at x = 1 is (1 - 1)·e^1 = 0
    [(x - 1)·e^x] at x = 0 is (0 - 1)·e^0 = -1
    
    Step 6: Subtract the lower value from the upper value.
    0 - (-1) = 1
    
    Step 7: Check the result.
    Differentiate the antiderivative: d/dx[(x - 1)·e^x] = e^x + (x - 1)·e^x = x·e^x, which matches the integrand.
    Numerical estimate: x·e^x is 0 at x = 0, about 0.824 at x = 0.5, and e ≈ 2.718 at x = 1. Simpson's rule gives (1/6)(0 + 4·0.824 + 2.718) = (1/6)(6.014) ≈ 1.002, which is close to 1.
    
    Final answer: 1

    454 tokens in, 519 out · 3.5 s · $0.0061 · 1 message on Pro · answered by anthropic/claude-sonnet-5.5 via Anthropic · September 28, 2026

12. Claude Sonnet 5

    Passed: Final answer 1: right.

    1. Use integration by parts: ∫x·e^x dx, let u = x, dv = e^x dx, so du = dx, v = e^x.
    
    2. Apply the formula ∫u dv = uv − ∫v du:
    ∫x·e^x dx = x·e^x − ∫e^x dx
    
    3. Compute the remaining integral:
    ∫e^x dx = e^x
    
    4. Combine results to get the antiderivative:
    ∫x·e^x dx = x·e^x − e^x + C = (x − 1)e^x + C
    
    5. Evaluate from 0 to 1 using the Fundamental Theorem of Calculus:
    F(1) = (1 − 1)e^1 = 0·e = 0
    F(0) = (0 − 1)e^0 = (−1)·1 = −1
    
    6. Subtract:
    F(1) − F(0) = 0 − (−1) = 1
    
    7. Check: Differentiate (x−1)e^x → derivative = e^x + (x−1)e^x = x·e^x, which matches the original integrand, confirming the antiderivative is correct.
    
    8. Sanity check by estimation: x·e^x is increasing and positive on [0,1], roughly averaging around (0 + e)/2 ≈ 1.36, times interval length 1 gives ≈1.36, but since the function is convex increasing weighted more near 1 slightly less than average estimate; exact value 1 is consistent with this rough range.
    
    Final answer: 1

    452 tokens in, 551 out (71 of them reasoning) · 6.4 s · $0.0064 · 1 message on Pro · answered by anthropic/claude-sonnet-5 via Anthropic · September 28, 2026

13. Mistral Large 4

    Passed: Final answer 1: right.

    Step 1: Identify the integration method.
    The integral involves a product of a polynomial (x) and an exponential function (e^x), so we use integration by parts.
    The formula is: integral of u dv = uv - integral of v du.
    
    Step 2: Choose u and dv.
    Let u = x and dv = e^x dx.
    This choice is good because differentiating u = x reduces it to a constant, simplifying the remaining integral.
    
    Step 3: Find du and v.
    du = dx
    v = e^x
    
    Step 4: Apply the integration by parts formula.
    integral of x·e^x dx = x·e^x - integral of e^x dx
    
    Step 5: Evaluate the remaining integral.
    integral of e^x dx = e^x
    
    Step 6: Write the antiderivative.
    integral of x·e^x dx = x·e^x - e^x + C
    Factoring out e^x: integral of x·e^x dx = e^x(x - 1) + C
    
    Step 7: Apply the limits of integration using the Fundamental Theorem of Calculus.
    integral from 0 to 1 of x·e^x dx = [e^x(x - 1)] from 0 to 1
    
    Step 8: Evaluate at the upper limit x = 1.
    e^1(1 - 1) = e · 0 = 0
    
    Step 9: Evaluate at the lower limit x = 0.
    e^0(0 - 1) = 1 · (-1) = -1
    
    Step 10: Subtract the lower limit evaluation from the upper limit evaluation.
    0 - (-1) = 1
    
    Step 11: Check the result by differentiating the antiderivative F(x) = e^x(x - 1).
    F'(x) = e^x(x - 1) + e^x(1) = x·e^x - e^x + e^x = x·e^x
    This matches the original integrand, so the antiderivative is correct.
    
    Final answer: 1

    349 tokens in, 3,043 out (2,019 of them reasoning) · 21.2 s · $0.0066 · 1 message on Pro · answered by mistralai/mistral-large-4-0 via Mistral · October 9, 2026

14. Kimi K3

    Passed: Final answer 1: right.

    We want to evaluate the definite integral of x·e^x from x = 0 to x = 1.
    
    Step 1: Choose integration by parts.
    The formula is: integral of u dv = u·v − integral of v du.
    Let u = x and dv = e^x dx.
    
    Step 2: Find du and v.
    du = dx (differentiate u = x).
    v = e^x (integrate dv = e^x dx).
    
    Step 3: Apply the formula to the indefinite integral.
    integral of x·e^x dx = x·e^x − integral of e^x dx
    = x·e^x − e^x + C
    = e^x·(x − 1) + C
    
    Step 4: Evaluate the antiderivative at the bounds.
    F(x) = e^x·(x − 1)
    F(1) = e^1·(1 − 1) = e·0 = 0
    F(0) = e^0·(0 − 1) = 1·(−1) = −1
    
    Step 5: Subtract to get the definite integral.
    integral from 0 to 1 of x·e^x dx = F(1) − F(0)
    = 0 − (−1)
    = 1
    
    Step 6: Check the result.
    Differentiate the antiderivative: d/dx [e^x·(x − 1)] = e^x·(x − 1) + e^x·1 = e^x·(x − 1 + 1) = x·e^x. This matches the integrand.
    Numerical check with Simpson's rule using x = 0, 1/2, 1: the estimate is (1/6)·(0 + 4·(1/2)·e^(1/2) + e) ≈ (1/6)·(3.297 + 2.718) ≈ 1.003, which is close to 1.
    
    Final answer: 1

    432 tokens in, 727 out (292 of them reasoning) · 64.4 s · $0.0069 · 1 message on Pro · answered by moonshotai/kimi-k3 via Wafer · September 28, 2026

15. Gemini 3.1 Pro

    Passed: Final answer 1: right.

    1. Identify the integral to evaluate.
    We need to find the definite integral of x*e^x dx from x = 0 to x = 1.
    
    2. Choose the method for integration.
    Because the function is a product of an algebraic term (x) and an exponential term (e^x), we use integration by parts. The formula is the integral of u dv = u*v - the integral of v du.
    
    3. Assign the parts u and dv.
    Let u = x, because its derivative becomes simpler.
    Let dv = e^x dx, because it is easy to integrate.
    
    4. Find du and v.
    Taking the derivative of u gives du = 1 dx.
    Integrating dv gives v = e^x.
    
    5. Apply the integration by parts formula to find the antiderivative.
    integral of x*e^x dx = x*e^x - integral of e^x dx
    Since the integral of e^x is just e^x, the antiderivative is:
    x*e^x - e^x
    
    6. Evaluate the antiderivative at the upper limit (x = 1).
    Substitute x = 1 into the expression:
    1*e^1 - e^1
    e - e = 0
    
    7. Evaluate the antiderivative at the lower limit (x = 0).
    Substitute x = 0 into the expression:
    0*e^0 - e^0
    0 - 1 = -1
    
    8. Subtract the lower limit value from the upper limit value.
    0 - (-1) = 1
    
    Check the result:
    Let's check the antiderivative by taking its derivative. Using the product rule on x*e^x, the derivative of (x*e^x - e^x) is:
    (1*e^x + x*e^x) - e^x
    e^x + x*e^x - e^x
    x*e^x
    This matches the original function inside the integral. Now, checking the arithmetic for the limits: at x = 1, it is 1*e - e = 0. At x = 0, it is 0 - 1 = -1. Subtracting these gives 0 - (-1) = 1. The result is verified and exact.
    
    Final answer: 1

    355 tokens in, 1,134 out (632 of them reasoning) · 9.6 s · $0.0072 · 1 message on Pro · answered by google/gemini-3.1-pro-preview via Google AI Studio · September 28, 2026

16. GPT-6 Astra

    Passed: Final answer 1: right.

    1. Use integration by parts: integral u dv = uv - integral v du.
    
    u = x, so du = dx.
    dv = e^x dx, so v = e^x.
    
    Therefore,
    integral x·e^x dx = x·e^x - integral e^x dx
    = x·e^x - e^x + C
    = (x - 1)e^x + C.
    
    2. Evaluate the antiderivative at the bounds and subtract.
    
    integral from 0 to 1 of x·e^x dx = [(x - 1)e^x] from 0 to 1
    = (1 - 1)e^1 - (0 - 1)e^0
    = 0 - (-1)
    = 1.
    
    3. Check by differentiating the antiderivative using the product rule.
    
    d/dx [(x - 1)e^x] = e^x + (x - 1)e^x
    = x·e^x.
    
    This matches the integrand, confirming the calculation.
    
    Final answer: 1

    346 tokens in, 230 out · 3.5 s · $0.0075 · 1 message on Pro · answered by openai/gpt-6-astra via OpenAI · September 28, 2026

17. Grok 4.7

    Passed: Final answer 1: right.

    The integral to evaluate is the definite integral of x * e^x from x = 0 to x = 1.
    
    Step 1: Find the antiderivative using integration by parts.
    Set u = x and dv = e^x dx.
    Then du = dx and v = e^x.
    The integration-by-parts formula gives
    x * e^x - integral of e^x dx.
    This simplifies to
    x * e^x - e^x + C,
    or equivalently
    e^x * (x - 1) + C.
    (The constant C is not needed for a definite integral.)
    
    Step 2: Differentiate the antiderivative to confirm it.
    d/dx [e^x * (x - 1)] = e^x * (x - 1) + e^x * 1
    = e^x * (x - 1 + 1)
    = x * e^x.
    The derivative matches the integrand, so the antiderivative is correct.
    
    Step 3: Evaluate the antiderivative at the limits.
    At x = 1:
    e^1 * (1 - 1) = 0.
    At x = 0:
    e^0 * (0 - 1) = -1.
    Subtract:
    0 - (-1) = 1.
    
    Step 4: Check the result another way.
    The series e^x = sum of x^n / n! implies that the integral equals sum of 1 / (n! * (n + 2)) from n = 0 to infinity.
    The partial sums are 1/2, then about 0.833, 0.958, 0.991, 0.998, and so on, approaching 1.
    This agrees with the exact evaluation.
    
    Final answer: 1

    1,584 tokens in, 1,278 out (901 of them reasoning) · 15.0 s · $0.0091 · 1 message on Pro · answered by x-ai/grok-4.7 via xAI · October 2, 2026

18. Claude Opus 5.5

    Passed: Final answer 1: right.

    We want the definite integral from 0 to 1 of x·e^x dx.
    
    1. Choose a method. The integrand is a product of x and e^x, so we use integration by parts: ∫u dv = u·v − ∫v du.
    
    2. Pick u and dv. Differentiating x makes it simpler, so let
       u = x, so du = dx
       dv = e^x dx, so v = e^x
    
    3. Apply the formula to get the antiderivative.
       ∫x·e^x dx = x·e^x − ∫e^x dx
       = x·e^x − e^x + C
       = (x − 1)·e^x + C
    
    4. Evaluate from 0 to 1 using the Fundamental Theorem of Calculus.
       At x = 1: (1 − 1)·e^1 = 0
       At x = 0: (0 − 1)·e^0 = −1
       Value = 0 − (−1) = 1
    
    5. Check by differentiating the antiderivative.
       d/dx [(x − 1)·e^x] = e^x + (x − 1)·e^x = x·e^x
       This matches the integrand, so the antiderivative is correct.
    
    6. Check with a rough estimate. The integrand x·e^x curves upward (it is convex) on [0, 1].
       Midpoint rule gives an underestimate: 1·(0.5·e^0.5) ≈ 0.82
       Trapezoid rule gives an overestimate: (0 + e)/2 ≈ 1.36
       The exact value 1 lies between these, as it should.
    
    Final answer: 1

    454 tokens in, 680 out (166 of them reasoning) · 6.2 s · $0.0154 · 1 message on Pro · answered by anthropic/claude-opus-5.5 via Google · September 28, 2026

19. Claude Fable 5.1

    Passed: Final answer 1: right.

    Problem: Evaluate the definite integral of x·e^x dx from x = 0 to x = 1.
    
    1. Choose a method. The integrand is a product of a polynomial (x) and an exponential (e^x), so integration by parts is the natural tool:
    integral of u dv = u·v − integral of v du.
    
    2. Pick u and dv. Let u = x so that it simplifies when differentiated, and let dv = e^x dx so that it is easy to integrate:
    u = x, so du = dx
    dv = e^x dx, so v = e^x
    
    3. Apply the formula to get the antiderivative:
    integral of x·e^x dx = x·e^x − integral of e^x dx
    = x·e^x − e^x + C
    = e^x(x − 1) + C
    
    4. Evaluate from 0 to 1 by substituting the limits into e^x(x − 1):
    At x = 1: e^1·(1 − 1) = e·0 = 0
    At x = 0: e^0·(0 − 1) = 1·(−1) = −1
    Subtract (upper minus lower): 0 − (−1) = 1
    
    5. Check by differentiating the antiderivative:
    d/dx [x·e^x − e^x] = (1·e^x + x·e^x) − e^x = x·e^x, which matches the integrand, so the antiderivative is correct.
    
    Second check by estimation: on [0, 1], x·e^x ranges from 0 to e ≈ 2.718 and is increasing and convex, so the integral should be a bit less than the trapezoid estimate (0 + 2.718)/2 ≈ 1.36. Simpson's rule with midpoint x = 0.5 gives (1/6)(0 + 4·0.5·e^0.5 + e) ≈ (1/6)(0 + 3.297 + 2.718) ≈ 1.003, very close to 1. Consistent.
    
    Final answer: 1

    454 tokens in, 624 out · 7.7 s · $0.0357 · 1 message on Pro · answered by anthropic/claude-fable-5.1 via Google · September 28, 2026

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