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Antiderivatives with initial conditions

Problem 3.378 · hard

Solve the initial-value problem \( \displaystyle f'(x) = 2 x^{2} - 3 \sin{\left(x \right)} \), \( \displaystyle f(1) = 4 \).

\[ \frac{d}{d x} \left(\frac{2 x^{3}}{3} + 3 \cos{\left(x \right)}\right) = 2 x^{2} - 3 \sin{\left(x \right)} \]
An antiderivative; every other differs by a constant C.✓ Proved
\[ 4 - \left. \frac{2 x^{3}}{3} + 3 \cos{\left(x \right)} \right|_{\substack{ x=1 }} = \frac{10}{3} - 3 \cos{\left(1 \right)} \]
f(1) = 4 fixes C.✓ Proved
\[ \frac{2 x^{3}}{3} + 3 \cos{\left(x \right)} - 3 \cos{\left(1 \right)} + \frac{10}{3} \]
The solution.✓ Proved

Answer \( f(x) = \frac{2 x^{3}}{3} + 3 \cos{\left(x \right)} - 3 \cos{\left(1 \right)} + \frac{10}{3} \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the antiderivative, calculates the constant of integration using the initial condition, and presents the final function. The algebraic steps are verified and the logic is sound.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the antiderivative, calculates the constant of integration using the initial condition, and presents the final function. The algebraic steps are verified and the logic is sound.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the antiderivative, solves for the constant of integration using the initial condition, and presents the correct final function.
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.