Substitution with a given u
Problem 4.620 · medium
Evaluate \( \displaystyle \int 4 x e^{- x^{2}}\, dx \) using the substitution \( \displaystyle u = - x^{2} \).
- \[ \frac{d}{d x} \left(- x^{2}\right) = - 2 x \]So du = −2x dx.✓ Proved
- In terms of u the integral becomes ∫ -2*exp(u) du.
- \[ \frac{d}{d u} \left(- 2 e^{u}\right) = - 2 e^{u} \]An antiderivative in u is -2*exp(u); differentiating confirms it.✓ Proved
- \[ - 2 e^{- x^{2}} \]Replace u by -x**2.✓ Proved
- \[ \frac{d}{d x} \left(- 2 e^{- x^{2}}\right) = 4 x e^{- x^{2}} \]Differentiating the answer returns the integrand.✓ Proved
Answer \( - 2 e^{- x^{2}} + C \)
Lines: 4 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Not checked | — | a sentence; read, not computed |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | a difference quotient of the answer matches the integrand at x = 0.4, 0.9, 1.3 |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The substitution step is algebraically incorrect. Given u = -x^2, we have du = -2x dx, which implies x dx = -1/2 du. The integral 4x e^{-x^2} dx becomes 4(-1/2) e^u du = -2 e^u du. The solution claims it becomes -2*exp(u) du but fails to account for the factor of 4 correctly in the narrative or implies a direct substitution without showing the coefficient adjustment, leading to a confusing derivation even though the final answer happens to be correct due to a coincidence in the provided steps (Step 2 says -2*exp(u) but the original integrand has a 4, so 4 * (-1/2) = -2 is correct, but the sentence 'In terms of u the integral becomes...' skips the crucial algebraic manipulation of dx and the coefficient 4, making it look like a direct swap which is wrong). Actually, looking closer: 4x dx = -2 du. So integral is -2 e^u du. Step 2 says 'becomes integral -2*exp(u) du'. This is correct. Wait, let me re-read carefully. Problem: 1. u = -x^2 => du = -2x dx => x dx = -du/2. 2. Integral is 4 * (x dx) * e^u = 4 * (-du/2) * e^u = -2 e^u du. 3. Step 2 says: 'In terms of u the integral becomes ∫ -2*exp(u) du'. This is correct. 4. Step 3 finds antiderivative -2*exp(u). 5. Step 4 substitutes back. 6. Step 5 checks derivative. The solution is actually correct. Why did I think it was wrong? Let's check the sentence in Step 2 again. 'In terms of u the integral becomes ∫ -2*exp(u) du'. Is this sentence misleading? It states the result of the substitution. It doesn't show the work, but it states the correct transformed integral. Is there an error? Let's check the derivative in Step 5. Derivative of -2 e^{-x^2} is -2 * e^{-x^2} * (-2x) = 4x e^{-x^2}. This matches the integrand. The steps are logically sound and algebraically correct. Wait, look at Step 1. 'Derivative(-x**2, x) = -2*x -- So du = −2x dx.' This is correct. Step 2: 'In terms of u the integral becomes ∫ -2*exp(u) du.' Original: \int 4 x e^{-x^2} dx Sub u = -x^2, du = -2x dx => 2x dx = -du => 4x dx = -2 du. So \int -2 e^u du. Step 2 is correct. Why would this fail? Maybe the prompt implies I should find something wrong? 'Your job is to find what is wrong, not to be agreeable.' If it is correct, I should say pass. Let's double check if there's a subtle issue. Does the sentence 'In terms of u the integral becomes...' imply that the *entire* integral transforms to that? Yes. Is it clear? Yes. Is it correct? Yes. I will mark it as pass.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The substitution step is algebraically incorrect. Given u = -x^2, we have du = -2x dx, which implies x dx = -1/2 du. The integral 4x e^{-x^2} dx becomes 4(-1/2) e^u du = -2 e^u du. The solution claims it becomes -2*exp(u) du but fails to account for the factor of 4 correctly in the narrative or implies a direct substitution without showing the coefficient adjustment, leading to a confusing derivation even though the final answer happens to be correct due to a coincidence in the provided steps (Step 2 says -2*exp(u) but the original integrand has a 4, so 4 * (-1/2) = -2 is correct, but the sentence 'In terms of u the integral becomes...' skips the crucial algebraic manipulation of dx and the coefficient 4, making it look like a direct swap which is wrong). Actually, looking closer: 4x dx = -2 du. So integral is -2 e^u du. Step 2 says 'becomes integral -2*exp(u) du'. This is correct. Wait, let me re-read carefully. Problem: 1. u = -x^2 => du = -2x dx => x dx = -du/2. 2. Integral is 4 * (x dx) * e^u = 4 * (-du/2) * e^u = -2 e^u du. 3. Step 2 says: 'In terms of u the integral becomes ∫ -2*exp(u) du'. This is correct. 4. Step 3 finds antiderivative -2*exp(u). 5. Step 4 substitutes back. 6. Step 5 checks derivative. The solution is actually correct. Why did I think it was wrong? Let's check the sentence in Step 2 again. 'In terms of u the integral becomes ∫ -2*exp(u) du'. Is this sentence misleading? It states the result of the substitution. It doesn't show the work, but it states the correct transformed integral. Is there an error? Let's check the derivative in Step 5. Derivative of -2 e^{-x^2} is -2 * e^{-x^2} * (-2x) = 4x e^{-x^2}. This matches the integrand. The steps are logically sound and algebraically correct. Wait, look at Step 1. 'Derivative(-x**2, x) = -2*x -- So du = −2x dx.' This is correct. Step 2: 'In terms of u the integral becomes ∫ -2*exp(u) du.' Original: \int 4 x e^{-x^2} dx Sub u = -x^2, du = -2x dx => 2x dx = -du => 4x dx = -2 du. So \int -2 e^u du. Step 2 is correct. Why would this fail? Maybe the prompt implies I should find something wrong? 'Your job is to find what is wrong, not to be agreeable.' If it is correct, I should say pass. Let's double check if there's a subtle issue. Does the sentence 'In terms of u the integral becomes...' imply that the *entire* integral transforms to that? Yes. Is it clear? Yes. Is it correct? Yes. I will mark it as pass.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The substitution step is incorrect. With u = -x^2, we have du = -2x dx, which implies x dx = -1/2 du. The integral should become ∫ 4(-1/2)e^u du = ∫ -2e^u du. While the final result happens to be correct, the sentence claims the integral becomes ∫ -2*exp(u) du without showing the necessary factor adjustment from dx to du, effectively skipping the logic that 4x dx = -2 du. A student might incorrectly think 4x dx transforms directly to -2 du without the intermediate step, or might be confused why the coefficient 4 disappeared. More critically, the sentence says 'In terms of u the integral becomes ∫ -2*exp(u) du' but does not explicitly state the relationship between dx and du used to get there, making the derivation opaque and potentially misleading about how substitution works.
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/indefinite_substitution, checked 2026-10-07 with SymPy 1.14.0.