Integral of \( \displaystyle e^{x + 1} \cos{\left(x + 1 \right)} \)
Problem 4.1283 · 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
- \[ = \int e e^{x} \cos{\left(x + 1 \right)}\, dx \]algebraUse the property exp(a+b) = exp(a)*exp(b).✓ Proved
- \[ = e \int e^{x} \cos{\left(x + 1 \right)}\, dx \]linearityFactor out the constant exp(1).✓ Proved
- \[ = e \int \left(- \sin{\left(1 \right)} \sin{\left(x \right)} + \cos{\left(1 \right)} \cos{\left(x \right)}\right) e^{x}\, dx \]trig-identityUse the cosine addition formula.✓ Proved
- \[ = - e \sin{\left(1 \right)} \int e^{x} \sin{\left(x \right)}\, dx + e \cos{\left(1 \right)} \int e^{x} \cos{\left(x \right)}\, dx \]linearity algebra simplifyDistribute the terms and factor out constants. Use integration by parts on the sine integral. Simplify the expression.✓ Proved
- \[ = e \left(e^{x} \cos{\left(x \right)} - \int \left(- e^{x} \sin{\left(x \right)}\right)\, dx\right) \cos{\left(1 \right)} - e \sin{\left(1 \right)} \int e^{x} \sin{\left(x \right)}\, dx \]partsApply integration by parts to the first integral.✓ Proved
- \[ = e e^{x} \cos{\left(1 \right)} \cos{\left(x \right)} - e \sin{\left(1 \right)} \int e^{x} \sin{\left(x \right)}\, dx + e \cos{\left(1 \right)} \int e^{x} \sin{\left(x \right)}\, dx \]partsEvaluate the integral of exp(x)*sin(x) using parts.✓ Proved
- \[ = e e^{x} \cos{\left(1 \right)} \cos{\left(x \right)} + \left(- e \sin{\left(1 \right)} + e \cos{\left(1 \right)}\right) \int e^{x} \sin{\left(x \right)}\, dx \]algebraGroup the terms involving the integral.✓ Proved
- \[ = \left(- e \sin{\left(1 \right)} + e \cos{\left(1 \right)}\right) \left(e^{x} \sin{\left(x \right)} - \int e^{x} \cos{\left(x \right)}\, dx\right) + e e^{x} \cos{\left(1 \right)} \cos{\left(x \right)} \]partsApply integration by parts to the remaining integral.✓ Proved
- \[ = \left(- e \sin{\left(1 \right)} + e \cos{\left(1 \right)}\right) e^{x} \sin{\left(x \right)} + e e^{x} \cos{\left(1 \right)} \cos{\left(x \right)} - \left(- e \sin{\left(1 \right)} + e \cos{\left(1 \right)}\right) \int e^{x} \cos{\left(x \right)}\, dx \]partsDistribute the terms.✓ Proved
- \[ = - \left(- e \sin{\left(1 \right)} + e \cos{\left(1 \right)}\right) \left(e^{x} \cos{\left(x \right)} - \int \left(- e^{x} \sin{\left(x \right)}\right)\, dx\right) + \left(- e \sin{\left(1 \right)} + e \cos{\left(1 \right)}\right) e^{x} \sin{\left(x \right)} + e e^{x} \cos{\left(1 \right)} \cos{\left(x \right)} \]partsSubstitute the integral of exp(x)*cos(x).✓ Proved
- \[ = \left(- e \sin{\left(1 \right)} + e \cos{\left(1 \right)}\right) e^{x} \sin{\left(x \right)} - \left(- e \sin{\left(1 \right)} + e \cos{\left(1 \right)}\right) e^{x} \cos{\left(x \right)} + e e^{x} \cos{\left(1 \right)} \cos{\left(x \right)} - \left(- e \sin{\left(1 \right)} + e \cos{\left(1 \right)}\right) \int e^{x} \sin{\left(x \right)}\, dx \]algebraExpand the expression.✓ Proved
- \[ = - \left(- e \sin{\left(1 \right)} + e \cos{\left(1 \right)}\right) \left(e^{x} \sin{\left(x \right)} - \int e^{x} \cos{\left(x \right)}\, dx\right) + \left(- e \sin{\left(1 \right)} + e \cos{\left(1 \right)}\right) e^{x} \sin{\left(x \right)} - \left(- e \sin{\left(1 \right)} + e \cos{\left(1 \right)}\right) e^{x} \cos{\left(x \right)} + e e^{x} \cos{\left(1 \right)} \cos{\left(x \right)} \]algebraWait, let's solve the linear equation for I = Integral(exp(x)cos(x),x) and J = Integral(exp(x)sin(x),x).✓ Proved
- \[ = - \left(- e \sin{\left(1 \right)} + e \cos{\left(1 \right)}\right) e^{x} \cos{\left(x \right)} + e e^{x} \cos{\left(1 \right)} \cos{\left(x \right)} + \left(- e \sin{\left(1 \right)} + e \cos{\left(1 \right)}\right) \int e^{x} \cos{\left(x \right)}\, dx \]algebraExpanding the integral term.✓ Proved
- \[ = \frac{e e^{x} \sin{\left(x + 1 \right)}}{2} + \frac{e e^{x} \cos{\left(x + 1 \right)}}{2} \]simplifySolving the system of integrals for exp(x)cos(x+1).✓ Proved
- \[ = \frac{\left(\sin{\left(x + 1 \right)} + \cos{\left(x + 1 \right)}\right) e^{x + 1}}{2} \]simplifyFinal simplification of the antiderivative.✓ Proved
Answer \( \frac{\sqrt{2} e^{x + 1} \sin{\left(x + \frac{\pi}{4} + 1 \right)}}{2} + C \)
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 |
| 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 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 15 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 16 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 17 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 18 | ✓ 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) — Several steps apply multiple rules at once or mislabel the rule used. For example, step 6 claims to use integration by parts but is labeled "algebra" and introduces an incorrect decomposition of the sine integral. Step 7 then simplifies an expression that was never correctly derived. The sequence of manipulations from steps 6 through 16 is mathematically incorrect and does not lead to a valid antiderivative. The solution therefore contains a serious error.qwen3.6:27b-mlx: fail (error) — The solution enters an infinite loop of applying integration by parts (steps 8-16) without ever solving the resulting linear system for the integrals. Step 17 abruptly jumps to the final answer without showing the algebraic resolution of the system, and the previous steps are logically incoherent as they do not reduce the problem.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-20 — The solution enters an infinite loop of applying integration by parts (steps 8-16) without ever solving the resulting linear system for the integrals. Step 17 abruptly jumps to the final answer without showing the algebraic resolution of the system, and the previous steps are logically incoherent as they do not reduce the problem.gpt-oss:20b: fail (error) 2026-10-20 — Several steps apply multiple rules at once or mislabel the rule used. For example, step 6 claims to use integration by parts but is labeled "algebra" and introduces an incorrect decomposition of the sine integral. Step 7 then simplifies an expression that was never correctly derived. The sequence of manipulations from steps 6 through 16 is mathematically incorrect and does not lead to a valid antiderivative. The solution therefore contains a serious error.qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution contains logical errors and invalid steps. Step 6 adds and subtracts the same integral, which is algebraically valid but labeled 'algebra' while the note claims 'integration by parts', creating a contradiction. More critically, Steps 8-15 attempt to solve for the integrals via repeated integration by parts but fail to actually solve the resulting linear system, instead just expanding terms indefinitely or incorrectly. Step 17 jumps to the final answer without showing the algebraic solution of the system of equations for the integrals, and the label 'simplify' is insufficient for solving a system of linear equations involving integrals. The transition from Step 16 to 17 is mathematically unjustified as written.gpt-oss:20b: fail (style) 2026-10-19 — Step 6 claims to use integration by parts but only performs an algebraic identity (adding and subtracting the same integral). The label "integration by parts" is therefore incorrect; it should be "algebra" or "rewrite". This mislabeling could mislead a student into thinking the step applied the product rule when it did not.qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution contains logical contradictions and circular reasoning. Step 6 adds and subtracts the same integral, which is algebraically valid but logically useless, and Step 7 immediately reverts to the state before Step 6, rendering the step redundant. Furthermore, Steps 8-15 attempt to solve the integrals via integration by parts but result in a circular expansion that does not isolate the unknowns, and Step 17 jumps to a final simplified form without showing the algebraic resolution of the system of equations implied in Step 15's note.gpt-oss:20b: fail (error) 2026-10-19 — Several steps apply multiple rules at once and contain incorrect algebraic manipulations, e.g., step 6 incorrectly rewrites the sine integral and step 7 simplifies an expression that is not equivalent. The solution therefore contains mathematical errors.
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 2026-10-20 with SymPy 1.14.0.