Integral of \( \displaystyle \sin{\left(2 x \right)} \cos{\left(x \right)} \)
Problem 4.177 · medium Video
Find \( \displaystyle \int \sin{\left(2 x \right)} \cos{\left(x \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \sin{\left(2 x \right)} \cos{\left(x \right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int 2 \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx \]substitution algebraUse the double angle identity sin(2x) = 2sin(x)cos(x). Simplify the expression.✓ Proved
- \[ = 2 \int \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx \]linearityMove the constant outside the integral.✓ Proved
- \[ = - \frac{2 \cos^{3}{\left(x \right)}}{3} \]antiderivative simplifyIntegrate using the power rule for the substitution u = cos(x). Simplify the final expression.✓ Proved
Answer \( - \frac{2 \cos^{3}{\left(x \right)}}{3} + C \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation. 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 (style) — Step 2 is labeled 'substitution' but applies a trigonometric identity (sin(2x) = 2sin(x)cos(x)); the correct label is 'trig-identity'. Step 5 is labeled 'antiderivative' but describes a u-substitution process; while the result is correct, the label 'substitution' would be more precise for the method described, though 'antiderivative' is acceptable if viewed as recognizing the form d/dx(cos^3(x)). However, the error in Step 2 is a clear mislabeling of the rule applied.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-03 — Step 2 is labeled 'substitution' but applies a trigonometric identity (sin(2x) = 2sin(x)cos(x)); the correct label is 'trig-identity'. Step 5 is labeled 'antiderivative' but describes a u-substitution process; while the result is correct, the label 'substitution' would be more precise for the method described, though 'antiderivative' is acceptable if viewed as recognizing the form d/dx(cos^3(x)). However, the error in Step 2 is a clear mislabeling of the rule applied.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (style) 2026-09-29 — Step 2 is labeled 'substitution' but applies a trigonometric identity (sin(2x) = 2sin(x)cos(x)); the correct label is 'trig-identity'. Step 5 is labeled 'antiderivative' but implicitly performs a u-substitution (u = cos(x)); while the result is correct, the label 'substitution' would be more precise for the method used, or 'antiderivative' is acceptable if viewed as recognizing the form d/dx(cos^3(x)), but given the explicit note about substitution, the label mismatch in Step 2 is the primary defect.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: fail (style) 2026-09-29 — Step 2 is labeled 'substitution' but applies a trigonometric identity (sin(2x) = 2sin(x)cos(x)); the correct label is 'trig-identity'. Step 5 is labeled 'antiderivative' but performs a u-substitution (u=cos(x)); while 'substitution' is in the vocabulary, 'antiderivative' usually implies recognizing a standard form directly, making the label imprecise given the note describes a substitution process.gpt-oss:20b: pass 2026-09-29
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.