Integral of \( \displaystyle \sin{\left(x \right)} \cos{\left(x \right)} \)
Problem 4.793 · easy
Find \( \displaystyle \int \sin{\left(x \right)} \cos{\left(x \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \sin{\left(x \right)} \cos{\left(x \right)}\, dx \]integralStart with the integral of the integrand.✓ Proved
- \[ = \int \frac{\sin{\left(2 x \right)}}{2}\, dx \]trig-identityUse the double angle identity sin(2x) = 2sin(x)cos(x).✓ Proved
- \[ = \frac{\int \sin{\left(2 x \right)}\, dx}{2} \]linearityFactor out the constant 1/2.✓ Proved
- \[ = - \frac{\cos{\left(2 x \right)}}{4} \]antiderivative simplifyIntegrate sin(2x) to get -cos(2x)/2. Simplify the expression.✓ Proved
Answer \( \frac{\sin^{2}{\left(x \right)}}{2} + C \)
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.
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 | lines differ by the constant 1/4 |
| 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 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: lines differ by the constant -1/4 |
| 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: pass
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly applies the double-angle identity, linearity, and standard antiderivatives. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.gpt-oss:20b: fail (error) 2026-10-10 — Step 4 applies two rules at once: it performs the antiderivative of sin(2x) and then multiplies by the outside constant 1/2. Each step must change only one thing, so this step should be split into two separate steps with appropriate labels.
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 2026-10-10 with SymPy 1.14.0.