Integral of \( \displaystyle \sin{\left(2 x \right)} \cos^{3}{\left(2 x \right)} \)
Problem 4.1338 · medium Video
Find \( \displaystyle \int \sin{\left(2 x \right)} \cos^{3}{\left(2 x \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \sin{\left(2 x \right)} \cos^{3}{\left(2 x \right)}\, dx \]integral algebra substitution simplifyStart with the integral of the given function. Commute the factors. Recognize the derivative of cos(2*x) is -2*sin(2*x). Move the constant outside the integral.✓ Proved
- \[ = - \frac{\cos^{4}{\left(2 x \right)}}{8} \]antiderivative simplifyIntegrate using the power rule for the substitution u = cos(2*x). Simplify the final expression.✓ Proved
Answer \( - \frac{\cos^{4}{\left(2 x \right)}}{8} + C \)
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 |
| 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 'Move the constant outside the integral' but the constant -1/2 remains inside the integral in the expression, and the label 'simplify' does not describe moving a constant out. Step 5 jumps from an integral expression to a result without showing the integration step or properly handling the constant factor, effectively applying the antiderivative rule and constant multiple rule simultaneously or incorrectly.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-25 — Step 4 claims to 'Move the constant outside the integral' but the constant -1/2 remains inside the integral in the expression, and the label 'simplify' does not describe moving a constant out. Step 5 jumps from an integral expression to a result without showing the integration step or properly handling the constant factor, effectively applying the antiderivative rule and constant multiple rule simultaneously or incorrectly.gpt-oss:20b: pass 2026-10-25qwen3.6:27b-mlx: fail (error) 2026-10-24 — Step 4 is labeled 'simplify' but performs the operation of pulling a constant factor out of the integral (linearity). Step 3 is labeled 'substitution' but merely rewrites sin(2x) in terms of the derivative of cos(2x); the actual substitution (change of variable) is not performed until Step 5, making the label in Step 3 inaccurate for the transformation shown.gpt-oss:20b: pass 2026-10-24qwen3.6:27b-mlx: fail (error) 2026-10-24 — Step 4 is labeled 'simplify' but performs the operation of pulling a constant factor out of the integral, which is the 'linearity' rule. Step 3 is labeled 'substitution' but merely rewrites sin(2x) in terms of the derivative of cos(2x); the actual substitution change of variables is not performed until Step 5, making the label in Step 3 inaccurate (it should be 'rewrite' or 'algebra').gpt-oss:20b: pass 2026-10-24
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 2026-10-25 with SymPy 1.14.0.