Integral of \( \displaystyle e^{2 x} \cos{\left(2 x \right)} \)
Problem 4.149 · medium Video
Find \( \displaystyle \int e^{2 x} \cos{\left(2 x \right)} \, dx \). (Omit the constant of integration.)
- \[ \int e^{2 x} \cos{\left(2 x \right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \frac{e^{2 x} \sin{\left(2 x \right)}}{2} - \int e^{2 x} \sin{\left(2 x \right)}\, dx \]partsUse integration by parts with u = exp(2*x) and dv = cos(2*x)dx.✓ Proved
- \[ = \frac{e^{2 x} \sin{\left(2 x \right)}}{2} + \frac{e^{2 x} \cos{\left(2 x \right)}}{2} + \int \left(- e^{2 x} \cos{\left(2 x \right)}\right)\, dx \]partsApply integration by parts again to the remaining integral.✓ Proved
- \[ = \frac{e^{2 x} \sin{\left(2 x \right)}}{2} + \frac{e^{2 x} \cos{\left(2 x \right)}}{2} - \int e^{2 x} \cos{\left(2 x \right)}\, dx \]simplifySimplify the expression by distributing the negative sign and simplifying the integrand.✓ Proved
- \[ = \frac{e^{2 x} \cos{\left(2 x \right)}}{2} + \int e^{2 x} \sin{\left(2 x \right)}\, dx \]parts simplify simplifyApply integration by parts to the remaining integral with u = exp(2*x) and dv = sin(2*x)dx. Distribute the negative sign. Combine like terms.✓ Proved
- \[ = - \int \left(- e^{2 x} \cos{\left(2 x \right)}\right)\, dx \]partsApply integration by parts to the remaining integral with u = exp(2*x) and dv = sin(2*x)dx.✓ Proved
- \[ = \int e^{2 x} \cos{\left(2 x \right)}\, dx \]simplify simplifySimplify the expression. The expression reduces back to the original integral.✓ Proved
- \[ = \frac{\left(\sin{\left(2 x \right)} + \cos{\left(2 x \right)}\right) e^{2 x}}{4} \]simplifySolving the loop or evaluating the parts leads to the final antiderivative.✓ Proved
Answer \( \frac{\sqrt{2} e^{2 x} \sin{\left(2 x + \frac{\pi}{4} \right)}}{4} + C \)
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. A reviewer's objection to the explanation stands; see 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 |
| 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 3 incorrectly handles the sign when applying integration by parts to ∫exp(2x)sin(2x)dx. The correct result should be -exp(2x)cos(2x)/2 + ∫exp(2x)cos(2x)dx, but the step produces a wrong sign for the integral term, leading to an incorrect subsequent simplification.qwen3.6:27b-mlx: fail (error) — The solution enters an infinite loop of integration by parts (Steps 2-10) without ever solving for the original integral algebraically. Step 11 jumps to the final answer without showing the necessary algebraic step of moving the integral term to the left-hand side and dividing by 2.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution enters an infinite loop of integration by parts (Steps 2-10) without ever solving for the original integral algebraically. Step 11 jumps to the final answer without showing the necessary algebraic step of moving the integral term to the left-hand side and dividing by 2.gpt-oss:20b: fail (error) 2026-10-03 — Step 3 incorrectly handles the sign when applying integration by parts to ∫exp(2x)sin(2x)dx. The correct result should be -exp(2x)cos(2x)/2 + ∫exp(2x)cos(2x)dx, but the step produces a wrong sign for the integral term, leading to an incorrect subsequent simplification.qwen3.6:27b-mlx: fail (error) 2026-09-29 — The solution enters an infinite loop of integration by parts (steps 2-10) without ever solving for the original integral algebraically. Step 11 jumps to the final answer without showing the necessary algebraic step of moving the integral term to the left-hand side and dividing by 2.gpt-oss:20b: fail (error) 2026-09-29 — Step 5 incorrectly applies integration by parts to the integral of exp(2*x)*cos(2*x) using dv = sin(2*x)dx, which does not match the integrand. This step changes more than one thing and mislabels the rule.qwen3.6:27b-mlx: fail (error) 2026-09-29 — The solution enters an infinite loop of integration by parts (Steps 2-10) without ever solving for the original integral algebraically. Step 11 jumps to the final answer without showing the necessary algebraic step of isolating the integral on one side of the equation.gpt-oss:20b: fail (error) 2026-09-29 — Step 3 applies integration by parts incorrectly: the sign of the resulting integral is reversed. This propagates through subsequent steps, leading to an incorrect antiderivative.
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 2026-10-03 with SymPy 1.14.0.