Integral of \( \displaystyle e^{x - 1} \sin{\left(e^{x - 1} \right)} \)
Problem 4.1245 · medium Video
Find \( \displaystyle \int e^{x - 1} \sin{\left(e^{x - 1} \right)} \, dx \). (Omit the constant of integration.)
- \[ \int e^{x - 1} \sin{\left(e^{x - 1} \right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \frac{e^{x} \sin{\left(\frac{e^{x}}{e} \right)}}{e}\, dx \]algebra algebraDistribute the exponent in the first term. Rearrange the terms using commutativity.✓ Proved
- \[ = \frac{\int e^{x} \sin{\left(\frac{e^{x}}{e} \right)}\, dx}{e} \]linearityPull the constant factor out of the integral.✓ Proved
- \[ = \frac{\int e^{x} \sin{\left(e^{x - 1} \right)}\, dx}{e} \]rewriteRewrite the argument of the sine function.✓ Proved
- \[ = \frac{\int e^{x} \sin{\left(\frac{e^{x}}{e} \right)}\, dx}{e} \]algebraExpand the exponent in the sine function argument.✓ Proved
- \[ = - \cos{\left(\frac{e^{x}}{e} \right)} \]substitution simplifyApply substitution u = exp(x) * exp(-1), where du = exp(x) * exp(-1) dx. Simplify the expression by canceling exp(-1).✓ Proved
- \[ = - \cos{\left(e^{x - 1} \right)} \]simplifySimplify the argument using the properties of exponents.✓ 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 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 9 | ✓ 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 (error) — Step 7 applies two rules at once: it performs the substitution u=exp(x)*exp(-1) and then algebraically divides by exp(-1). The correct substitution yields exp(-1)*(-cos(u)), not exp(-1)*(-cos(u)/exp(-1)).qwen3.6:27b-mlx: fail (error) — Step 7 claims to use substitution u = exp(x - 1), but the note states du = exp(x) * exp(-1) dx, which is incorrect; the derivative of exp(x - 1) is exp(x - 1) dx. The step effectively performs the integration correctly but provides a mathematically false justification for the differential relationship.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — Step 7 claims to use substitution u = exp(x - 1), but the note states du = exp(x) * exp(-1) dx, which is incorrect; the derivative of exp(x - 1) is exp(x - 1) dx. The step effectively performs the integration correctly but provides a mathematically false justification for the differential relationship.gpt-oss:20b: fail (error) 2026-10-19 — Step 7 applies two rules at once: it performs the substitution u=exp(x)*exp(-1) and then algebraically divides by exp(-1). The correct substitution yields exp(-1)*(-cos(u)), not exp(-1)*(-cos(u)/exp(-1)).qwen3.6:27b-mlx: fail (error) 2026-10-16 — Step 7 is mathematically incorrect. The substitution u = exp(x-1) implies du = exp(x-1) dx, so the integral of exp(x-1)sin(exp(x-1)) dx is -cos(exp(x-1)). However, the step claims the result is -cos(exp(x-1))/exp(-1), which equals -e*cos(exp(x-1)). This introduces an erroneous factor of e. The previous steps pulled out exp(-1) (1/e), so the final result in step 8 becomes -e * (1/e) * cos(...) = -cos(...), which accidentally matches the correct answer, but the intermediate step 7 contains a false equality (the integral was not divided by exp(-1)). Specifically, Integral(exp(x)sin(exp(x-1))dx) = -cos(exp(x-1)), not -cos(exp(x-1))/exp(-1).gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-16 — Step 7 claims the substitution is u = exp(x) * exp(-1) with du = exp(x) * exp(-1) dx, but the correct differential is du = exp(x) dx. The step incorrectly includes the constant factor exp(-1) in the differential, which contradicts the algebraic setup in steps 4-6 where exp(-1) was factored out.gpt-oss:20b: fail (style) 2026-10-16 — Step 7 applies two rules at once: it performs the substitution u=exp(x)*exp(-1) and simultaneously simplifies the resulting expression by dividing by exp(-1). According to the contract, each step must change only one thing, so this step should be split into separate substitution and algebraic simplification steps.
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 2026-10-19 with SymPy 1.14.0.