Integral of \( \displaystyle e^{3 x + 1} \sin{\left(e^{3 x + 1} \right)} \)
Problem 4.923 · medium
Find \( \displaystyle \int e^{3 x + 1} \sin{\left(e^{3 x + 1} \right)} \, dx \). (Omit the constant of integration.)
- \[ \int e^{3 x + 1} \sin{\left(e^{3 x + 1} \right)}\, dx \]integral algebra rewrite algebraStart with the integral of the given function. Multiply and divide by 3 to prepare for substitution. Rewrite the integrand to highlight the inner function's derivative. Simplify the expression back to a standard form.✓ Proved
- \[ = - \frac{\cos{\left(e^{3 x + 1} \right)}}{3} \]antiderivativeIntegrate using the substitution u = exp(3*x + 1).✓ Proved
Answer \( - \frac{\cos{\left(e^{3 x + 1} \right)}}{3} + C \)
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 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated the stated antiderivative back to the integrand |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — Step 3 introduces a `diff` operator into the integrand, which is not part of the standard algebraic manipulation allowed by the 'rewrite' rule; furthermore, Step 4 simplifies `diff(exp(3*x + 1), x)` back to `3*exp(3*x + 1)` using 'algebra', but 'algebra' does not cover differentiation. The transition from Step 4 to Step 5 performs the actual integration (substitution) but is labeled 'antiderivative' without showing the substitution step explicitly as a separate line, violating the one-change-per-step rule if we consider the substitution logic implicit, or more critically, Step 3 and 4 are logically redundant and misuse labels to hide the differentiation step.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (error) 2026-10-12 — Step 3 introduces a `diff` operator into the integrand, which is not part of the standard algebraic manipulation allowed by the 'rewrite' rule; furthermore, Step 4 simplifies `diff(exp(3*x + 1), x)` back to `3*exp(3*x + 1)` using 'algebra', but 'algebra' does not cover differentiation. The transition from Step 4 to Step 5 performs the actual integration (substitution) but is labeled 'antiderivative' without showing the substitution step explicitly as a separate line, violating the one-change-per-step rule if we consider the substitution logic implicit, or more critically, Step 3 and 4 are logically redundant and misuse labels to hide the differentiation step.qwen3.6:27b-mlx: fail (error) 2026-10-12 — Step 4 is mathematically incorrect; it simplifies 3*exp(3*x + 1)/3 to exp(3*x + 1), but the integrand in Step 3 was sin(exp(3*x + 1))*3*exp(3*x + 1)/3, so the sine term must remain. Step 4 drops the sine term entirely, changing the integrand from exp(3*x + 1)*sin(exp(3*x + 1)) to just exp(3*x + 1).gpt-oss:20b: pass 2026-10-12
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
gemma4:26b, checked 2026-10-12 with SymPy 1.14.0.