Integral of \( \displaystyle e^{x - 1} \cos{\left(x - 1 \right)} \)
Problem 4.1312 · medium Video
Find \( \displaystyle \int e^{x - 1} \cos{\left(x - 1 \right)} \, dx \). (Omit the constant of integration.)
- \[ \int e^{x - 1} \cos{\left(x - 1 \right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \left(\sin{\left(1 \right)} \sin{\left(x \right)} + \cos{\left(1 \right)} \cos{\left(x \right)}\right) e^{x - 1}\, dx \]trig-identityUse the angle subtraction formula for cosine.✓ Proved
- \[ = \sin{\left(1 \right)} \int e^{x - 1} \sin{\left(x \right)}\, dx + \cos{\left(1 \right)} \int e^{x - 1} \cos{\left(x \right)}\, dx \]linearityDistribute the integral over the sum.✓ Proved
- \[ = \int e^{x - 1} \cos{\left(x - 1 \right)}\, dx \]simplifyThe previous expansion steps were redundant; we can evaluate the integral directly.✓ Proved
- \[ = \frac{\left(\sin{\left(x - 1 \right)} + \cos{\left(x - 1 \right)}\right) e^{x - 1}}{2} \]antiderivativeApply the standard formula for the integral of e^ax * cos(bx).✓ Proved
Answer \( \frac{\sqrt{2} e^{x - 1} \cos{\left(- x + \frac{\pi}{4} + 1 \right)}}{2} + C \)
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 4 rewrites the expression from a sum of two integrals back to a single integral, but labels it "simplify". The correct label for this operation would be "rewrite" (or possibly "linearity" if the step were a direct application of that rule).qwen3.6:27b-mlx: fail (error) — Step 4 reverts the expression to the original integral, creating a logical loop that invalidates the subsequent application of the antiderivative rule in Step 5. The solution fails to actually perform the integration steps implied by the transition from Step 3 to Step 5.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-22 — Step 4 reverts the expression to the original integral, creating a logical loop that invalidates the subsequent application of the antiderivative rule in Step 5. The solution fails to actually perform the integration steps implied by the transition from Step 3 to Step 5.gpt-oss:20b: fail (style) 2026-10-22 — Step 4 rewrites the expression from a sum of two integrals back to a single integral, but labels it "simplify". The correct label for this operation would be "rewrite" (or possibly "linearity" if the step were a direct application of that rule).qwen3.6:27b-mlx: fail (error) 2026-10-21 — Step 4 reverts the expression to the original integral, creating a logical loop that ignores the work done in steps 2 and 3. Step 5 then claims to apply a standard formula to the original integral, but the transition from step 4 to 5 skips the actual derivation or justification of the antiderivative, effectively asserting the answer without showing the integration steps (like integration by parts) required to reach it.gpt-oss:20b: pass 2026-10-21qwen3.6:27b-mlx: fail (error) 2026-10-21 — Step 4 undoes the work of steps 2 and 3, creating a logical loop where the problem is restated as its own solution without justification. Step 5 applies the antiderivative rule to the original integrand but fails to account for the constant shift in the exponent and argument, resulting in an incorrect final expression (the correct antiderivative involves exp(x-1)cos(x-1) and exp(x-1)sin(x-1), but the coefficients and structure in the stated answer are derived from an invalid step sequence).gpt-oss:20b: fail (error) 2026-10-21 — Step 4 replaces the sum of two separate integrals with the original integral without applying a valid rule. This changes more than one thing at once and is not justified by any of the allowed labels.
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 2026-10-22 with SymPy 1.14.0.