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Antiderivatives with initial conditions

Problem 3.580 · medium

Solve the initial-value problem \( \displaystyle f'(x) = 2 x^{3} + \sin{\left(x \right)} \), \( \displaystyle f(0) = 0 \).

\[ \frac{d}{d x} \left(\frac{x^{4}}{2} - \cos{\left(x \right)}\right) = 2 x^{3} + \sin{\left(x \right)} \]
An antiderivative; every other differs by a constant C.✓ Proved
\[ - \left. \frac{x^{4}}{2} - \cos{\left(x \right)} \right|_{\substack{ x=0 }} = 1 \]
f(0) = 0 fixes C.✓ Proved
\[ \frac{x^{4}}{2} - \cos{\left(x \right)} + 1 \]
The solution.✓ Proved

Answer \( f(x) = \frac{x^{4}}{2} - \cos{\left(x \right)} + 1 \)

Every line of this solution was proved by the computer algebra system SymPy. 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	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ 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	the answer differentiated back to the given derivative, and the initial values checked

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution incorrectly determines the constant of integration. Since the particular antiderivative F(x) = x^4/2 - cos(x) satisfies F(0) = -1, and the initial condition is f(0) = 0, the constant C must be 1 (so f(x) = F(x) + 1). However, the text claims 'f(0) = 0 fixes C' after showing '-Subs(...) = 1', which is confusingly phrased and mathematically imprecise: it implies the substitution result *is* the fix, rather than showing the calculation C = f(0) - F(0). More critically, line 2 shows '-Subs(...) = 1', which evaluates to -(-1) = 1, but the logic connecting this to C=1 is opaque and potentially misleading if interpreted as C = -1. Actually, looking closer: F(0) = -1. We need f(0) = F(0) + C = 0 => -1 + C = 0 => C = 1. The solution adds 1 in line 3. The error is in the explanation of line 2. It says '-Subs(...) = 1'. This is true. But it says 'f(0)=0 fixes C'. It doesn't explicitly state C=1. It just jumps to the final answer. Is it an error? The final answer is correct. The intermediate step is just a check. Wait, line 2 is marked 'proved'. The text says '-Subs(...) = 1'. This is just a calculation. The sentence 'f(0)=0 fixes C' is the reasoning. The reasoning is incomplete but not false. However, usually, one writes C = 0 - F(0). Here it seems to imply C is derived from that value. Let's look at the 'Stated answer'. It is correct. The solution steps are: 1. Find general antiderivative. 2. Evaluate particular part at 0. 3. Write final answer. The jump from 2 to 3 assumes the reader knows C = -F(0). This is standard. Is there an error? No, the math is correct. The phrasing is terse. Is it 'misleading'? A student might think C = 1 because the substitution gave 1? No, F(0)=-1. -F(0)=1. So C=1. It's correct. Why did I think it was an error? Let me re-read carefully. Line 2: `-Subs(x**4/2 - cos(x), x, 0) = 1`. This calculates `-F(0)`. Since `f(x) = F(x) + C`, `f(0) = F(0) + C`. `0 = F(0) + C` => `C = -F(0)`. So `C = 1`. The line calculates `C`. The sentence says `f(0)=0 fixes C`. This is correct. The solution is sound.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution incorrectly determines the constant of integration. Since the particular antiderivative F(x) = x^4/2 - cos(x) satisfies F(0) = -1, and the initial condition is f(0) = 0, the constant C must be 1 (so f(x) = F(x) + 1). However, the text claims 'f(0) = 0 fixes C' after showing '-Subs(...) = 1', which is confusingly phrased and mathematically imprecise: it implies the substitution result *is* the fix, rather than showing the calculation C = f(0) - F(0). More critically, line 2 shows '-Subs(...) = 1', which evaluates to -(-1) = 1, but the logic connecting this to C=1 is opaque and potentially misleading if interpreted as C = -1. Actually, looking closer: F(0) = -1. We need f(0) = F(0) + C = 0 => -1 + C = 0 => C = 1. The solution adds 1 in line 3. The error is in the explanation of line 2. It says '-Subs(...) = 1'. This is true. But it says 'f(0)=0 fixes C'. It doesn't explicitly state C=1. It just jumps to the final answer. Is it an error? The final answer is correct. The intermediate step is just a check. Wait, line 2 is marked 'proved'. The text says '-Subs(...) = 1'. This is just a calculation. The sentence 'f(0)=0 fixes C' is the reasoning. The reasoning is incomplete but not false. However, usually, one writes C = 0 - F(0). Here it seems to imply C is derived from that value. Let's look at the 'Stated answer'. It is correct. The solution steps are: 1. Find general antiderivative. 2. Evaluate particular part at 0. 3. Write final answer. The jump from 2 to 3 assumes the reader knows C = -F(0). This is standard. Is there an error? No, the math is correct. The phrasing is terse. Is it 'misleading'? A student might think C = 1 because the substitution gave 1? No, F(0)=-1. -F(0)=1. So C=1. It's correct. Why did I think it was an error? Let me re-read carefully. Line 2: `-Subs(x**4/2 - cos(x), x, 0) = 1`. This calculates `-F(0)`. Since `f(x) = F(x) + C`, `f(0) = F(0) + C`. `0 = F(0) + C` => `C = -F(0)`. So `C = 1`. The line calculates `C`. The sentence says `f(0)=0 fixes C`. This is correct. The solution is sound.
gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: inconclusive 2026-10-11 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly determines the constant of integration. The particular antiderivative F(x) = x^4/2 - cos(x) satisfies F(0) = -1. To satisfy
gpt-oss:20b: pass 2026-10-11

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/antiderivative_ivp, checked 2026-10-11 with SymPy 1.14.0.