Integral of \( \displaystyle e^{x + 1} \sin{\left(e^{x + 1} \right)} \)
Problem 4.1341 · medium Video
Find \( \displaystyle \int e^{x + 1} \sin{\left(e^{x + 1} \right)} \, dx \). (Omit the constant of integration.)
- \[ \int e^{x + 1} \sin{\left(e^{x + 1} \right)}\, dx \]integral rewrite substitution substitutionStart with the integral of the given function. Rewrite the integrand to prepare for substitution. This step is conceptual; we substitute exp(x + 1) with a dummy variable, but since we cannot use 'u', we perform the substitution by recognizing the derivative of the inner function is present. Recognize that the integrand is in the form f(g(x)) * g'(x).✓ Proved
- \[ = - \cos{\left(e^{x + 1} \right)} \]antiderivativeThe antiderivative of sin(exp(x + 1)) * exp(x + 1) is -cos(exp(x + 1)).✓ Proved
Answer \( - \cos{\left(e^{x + 1} \right)} + C \)
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. How we verify
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: fail (style) — Step 3 combines two separate operations (adding and subtracting the same integral) in a single step, violating the rule that each step must change only one thing. This unnecessary manipulation also mislabels the operation as a substitution when no actual substitution is performed.qwen3.6:27b-mlx: fail (error) — Step 3 is mathematically nonsensical, adding and subtracting the same integral without performing any actual substitution or transformation. Step 4 repeats the integrand from Step 2 without change, failing to demonstrate the substitution process.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-26 — Step 3 is mathematically nonsensical, adding and subtracting the same integral without performing any actual substitution or transformation. Step 4 repeats the integrand from Step 2 without change, failing to demonstrate the substitution process.gpt-oss:20b: fail (style) 2026-10-26 — Step 3 combines two separate operations (adding and subtracting the same integral) in a single step, violating the rule that each step must change only one thing. This unnecessary manipulation also mislabels the operation as a substitution when no actual substitution is performed.qwen3.6:27b-mlx: fail (error) 2026-10-25 — Step 3 is mathematically nonsensical; it adds and subtracts the same integral, which is not a valid substitution step. Step 4 then claims to perform a substitution that was not actually executed in the previous step, violating the one-step-one-rule contract and logical continuity.gpt-oss:20b: fail (style) 2026-10-25 — Step 3 adds and subtracts the same integral in one line, changing two things at once, and labels it "substitution" even though it is merely an algebraic manipulation. This violates the one‑change‑per‑step rule and mislabels the operation.qwen3.6:27b-mlx: fail (error) 2026-10-25 — Step 3 is mathematically nonsensical, adding and subtracting the same integral without changing the expression, and misuses the 'substitution' label for a step that performs no actual substitution. Step 4 also mislabels the recognition of the chain rule pattern as 'substitution' when no variable change occurs.gpt-oss:20b: fail (style) 2026-10-25 — Step 3 applies two operations (adding and subtracting the same integral) in one line and labels it "substitution" even though no substitution is performed. This violates the rule that each step must change only one thing and use an appropriate label.
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 2026-10-26 with SymPy 1.14.0.