Integral of \( \displaystyle e^{2 x} \sin{\left(2 x \right)} \)
Problem 4.1399 · medium Video
Find \( \displaystyle \int e^{2 x} \sin{\left(2 x \right)} \, dx \). (Omit the constant of integration.)
- \[ \int e^{2 x} \sin{\left(2 x \right)}\, dx \]integralStart with the integral.✓ Proved
- \[ = - \frac{e^{2 x} \cos{\left(2 x \right)}}{2} + \int e^{2 x} \cos{\left(2 x \right)}\, dx \]partsUse integration by parts with u = exp(2*x) and dv = sin(2*x) dx.✓ Proved
- \[ = \frac{e^{2 x} \sin{\left(2 x \right)}}{2} - \frac{e^{2 x} \cos{\left(2 x \right)}}{2} - \int e^{2 x} \sin{\left(2 x \right)}\, dx \]partsApply integration by parts to the remaining integral with u = exp(2*x) and dv = cos(2*x) dx.✓ Proved
- \[ = \frac{e^{2 x} \sin{\left(2 x \right)}}{4} - \frac{e^{2 x} \cos{\left(2 x \right)}}{4} \]simplifySolve the equation for the integral.✓ Proved
Answer \( - \frac{\sqrt{2} e^{2 x} \cos{\left(2 x + \frac{\pi}{4} \right)}}{4} + C \)
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. 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 |
| 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: passqwen3.6:27b-mlx: fail (error) — Step 4 claims to use 'simplify' but actually performs algebraic manipulation to solve for the original integral (moving the integral term to the left side and dividing by 2). This is a logical step of solving an equation, not a simplification, and the label 'simplify' does not capture the operation performed. Furthermore, the result in Step 4 is missing the factor of 1/2 that should arise from dividing by (1+1) or similar, wait, let's re-verify the math. Let I = Integral(exp(2x)sin(2x)dx). Step 2: u=e^{2x}, dv=sin(2x)dx -> du=2e^{2x}dx, v=-cos(2x)/2. I = -e^{2x}cos(2x)/2 - Integral(-cos(2x)/2 * 2e^{2x}dx) = -e^{2x}cos(2x)/2 + Integral(e^{2x}cos(2x)dx). Step 3: J = Integral(e^{2x}cos(2x)dx). u=e^{2x}, dv=cos(2x)dx -> du=2e^{2x}dx, v=sin(2x)/2. J = e^{2x}sin(2x)/2 - Integral(sin(2x)/2 * 2e^{2x}dx) = e^{2x}sin(2x)/2 - I. So I = -e^{2x}cos(2x)/2 + e^{2x}sin(2x)/2 - I. 2I = e^{2x}(sin(2x) - cos(2x))/2. I = e^{2x}(sin(2x) - cos(2x))/4. Step 4 result: exp(2*x)*sin(2*x)/4 - exp(2*x)*cos(2*x)/4. This matches. The error is purely in the labeling. Step 4 is not 'simplify'. It is 'algebra' (solving for the variable). The contract says 'algebra' is a valid label. 'Simplify' usually implies reducing complexity without changing the form's structural role (like combining fractions). Solving for I is a distinct logical step. Is it a defect? Yes, 'simplify' is a misnomer for 'solve for the integral'. Also, the final answer provided in the task description is `-sqrt(2)*exp(2*x)*cos(2*x + pi/4)/4`. Let's check if Step 4's result equals this. `sin(2x) - cos(2x) = sqrt(2) * (1/sqrt(2)sin(2x) - 1/sqrt(2)cos(2x)) = sqrt(2) * (sin(2x)cos(pi/4) - cos(2x)sin(pi/4)) = sqrt(2)sin(2x - pi/4)`. Also `sin(2x) - cos(2x) = -sqrt(2) * (cos(2x) - sin(2x)) = -sqrt(2) * (cos(2x)cos(pi/4) + sin(2x)sin(pi/4)) = -sqrt(2)cos(2x - pi/4) = -sqrt(2)cos(2x + 3pi/4)`? No. `cos(A+B) = cosA cosB - sinA sinB`. `cos(2x + pi/4) = cos(2x)/sqrt(2) - sin(2x)/sqrt(2)`. So `-sqrt(2)cos(2x+pi/4) = -cos(2x) + sin(2x)`. Yes. So the final answer is correct. The defect is the label in Step 4. It calls a step that solves for the integral 'simplify'. This is misleading/wrong labeling. The rule applied is algebraic manipulation to isolate the term. 'Algebra' is the correct label. 'Simplify' is not.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-31 — Step 4 claims to use 'simplify' but actually performs algebraic manipulation to solve for the original integral (moving the integral term to the left side and dividing by 2). This is a logical step of solving an equation, not a simplification, and the label 'simplify' does not capture the operation performed. Furthermore, the result in Step 4 is missing the factor of 1/2 that should arise from dividing by (1+1) or similar, wait, let's re-verify the math. Let I = Integral(exp(2x)sin(2x)dx). Step 2: u=e^{2x}, dv=sin(2x)dx -> du=2e^{2x}dx, v=-cos(2x)/2. I = -e^{2x}cos(2x)/2 - Integral(-cos(2x)/2 * 2e^{2x}dx) = -e^{2x}cos(2x)/2 + Integral(e^{2x}cos(2x)dx). Step 3: J = Integral(e^{2x}cos(2x)dx). u=e^{2x}, dv=cos(2x)dx -> du=2e^{2x}dx, v=sin(2x)/2. J = e^{2x}sin(2x)/2 - Integral(sin(2x)/2 * 2e^{2x}dx) = e^{2x}sin(2x)/2 - I. So I = -e^{2x}cos(2x)/2 + e^{2x}sin(2x)/2 - I. 2I = e^{2x}(sin(2x) - cos(2x))/2. I = e^{2x}(sin(2x) - cos(2x))/4. Step 4 result: exp(2*x)*sin(2*x)/4 - exp(2*x)*cos(2*x)/4. This matches. The error is purely in the labeling. Step 4 is not 'simplify'. It is 'algebra' (solving for the variable). The contract says 'algebra' is a valid label. 'Simplify' usually implies reducing complexity without changing the form's structural role (like combining fractions). Solving for I is a distinct logical step. Is it a defect? Yes, 'simplify' is a misnomer for 'solve for the integral'. Also, the final answer provided in the task description is `-sqrt(2)*exp(2*x)*cos(2*x + pi/4)/4`. Let's check if Step 4's result equals this. `sin(2x) - cos(2x) = sqrt(2) * (1/sqrt(2)sin(2x) - 1/sqrt(2)cos(2x)) = sqrt(2) * (sin(2x)cos(pi/4) - cos(2x)sin(pi/4)) = sqrt(2)sin(2x - pi/4)`. Also `sin(2x) - cos(2x) = -sqrt(2) * (cos(2x) - sin(2x)) = -sqrt(2) * (cos(2x)cos(pi/4) + sin(2x)sin(pi/4)) = -sqrt(2)cos(2x - pi/4) = -sqrt(2)cos(2x + 3pi/4)`? No. `cos(A+B) = cosA cosB - sinA sinB`. `cos(2x + pi/4) = cos(2x)/sqrt(2) - sin(2x)/sqrt(2)`. So `-sqrt(2)cos(2x+pi/4) = -cos(2x) + sin(2x)`. Yes. So the final answer is correct. The defect is the label in Step 4. It calls a step that solves for the integral 'simplify'. This is misleading/wrong labeling. The rule applied is algebraic manipulation to isolate the term. 'Algebra' is the correct label. 'Simplify' is not.gpt-oss:20b: pass 2026-10-31qwen3.6:27b-mlx: fail (error) 2026-10-30 — Step 4 claims to use 'simplify' to solve the equation for the integral, but solving an equation is not a simplification step; it requires algebraic manipulation (collecting terms and dividing) which is not captured by the label 'simplify'. Furthermore, the result in Step 4 is missing the constant of integration handling or the final division by 2 required to isolate the integral, making the mathematical transition from Step 3 to Step 4 incorrect as labeled and executed.gpt-oss:20b: pass 2026-10-30qwen3.6:27b-mlx: fail (error) 2026-10-30 — Step 4 claims to use 'simplify' but actually performs algebraic manipulation to solve for the original integral (moving terms and dividing by 2), which is not a simplification step. Additionally, the result in Step 4 is missing the factor of 1/2 that arises from solving I = -e^{2x}cos(2x)/2 + e^{2x}sin(2x)/2 - I, leading to 2I = ..., so I = .../2. The stated answer also differs from the standard form by a constant factor or phase shift error if not carefully checked, but the primary defect is the mislabeled and mathematically incorrect transition in step 4.gpt-oss:20b: pass 2026-10-30
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 2026-10-31 with SymPy 1.14.0.