Antiderivatives with initial conditions
Problem 3.380 · medium
Solve the initial-value problem \( \displaystyle f'(x) = 3 x^{2} - 3 \cos{\left(x \right)} \), \( \displaystyle f(0) = -3 \).
- \[ \frac{d}{d x} \left(x^{3} - 3 \sin{\left(x \right)}\right) = 3 x^{2} - 3 \cos{\left(x \right)} \]An antiderivative; every other differs by a constant C.✓ Proved
- \[ - \left. x^{3} - 3 \sin{\left(x \right)} \right|_{\substack{ x=0 }} - 3 = -3 \]f(0) = -3 fixes C.✓ Proved
- \[ x^{3} - 3 \sin{\left(x \right)} - 3 \]The solution.✓ Proved
Answer \( f(x) = x^{3} - 3 \sin{\left(x \right)} - 3 \)
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: fail (misleading) — The solution omits the constant of integration in the antiderivative, yet claims that f(0) = –3 fixes C. A student may incorrectly assume the antiderivative is unique without adding the constant term.qwen3.6:27b-mlx: pass — The solution correctly identifies the general antiderivative and uses the initial condition to determine the constant of integration. The algebraic verification steps are correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the general antiderivative and uses the initial condition to determine the constant of integration. The algebraic verification steps are correct.gpt-oss:20b: fail (misleading) 2026-10-04 — The solution omits the constant of integration in the antiderivative, yet claims that f(0) = –3 fixes C. A student may incorrectly assume the antiderivative is unique without adding the constant term.qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the general antiderivative and applies the initial condition to determine the constant of integration. The algebraic steps are valid and the final answer is correct.gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.