Antiderivatives with initial conditions
Problem 3.525 · medium
Solve the initial-value problem \( \displaystyle f'(x) = 4 - \cos{\left(x \right)} \), \( \displaystyle f(1) = 2 \).
- \[ \frac{d}{d x} \left(4 x - \sin{\left(x \right)}\right) = 4 - \cos{\left(x \right)} \]An antiderivative; every other differs by a constant C.✓ Proved
- \[ 2 - \left. 4 x - \sin{\left(x \right)} \right|_{\substack{ x=1 }} = -2 + \sin{\left(1 \right)} \]f(1) = 2 fixes C.✓ Proved
- \[ 4 x - \sin{\left(x \right)} - 2 + \sin{\left(1 \right)} \]The solution.✓ Proved
Answer \( f(x) = 4 x - \sin{\left(x \right)} - 2 + \sin{\left(1 \right)} \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.
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: passqwen3.6:27b-mlx: pass — The solution correctly identifies the general antiderivative, solves for the constant of integration using the initial condition, and presents the correct final function.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly identifies the general antiderivative, solves for the constant of integration using the initial condition, and presents the correct final function.gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly identifies the general antiderivative, solves for the constant of integration using the initial condition, and presents the final result. The algebraic steps are verified.
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 2026-10-09 with SymPy 1.14.0.