Integral of \( \displaystyle \cos^{2}{\left(3 x + 1 \right)} \)
Problem 4.274 · medium
Find \( \displaystyle \int \cos^{2}{\left(3 x + 1 \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \cos^{2}{\left(3 x + 1 \right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \left(\frac{\cos{\left(6 x + 2 \right)}}{2} + \frac{1}{2}\right)\, dx \]trig-identityUse the power-reduction identity cos(u)^2 = (1 + cos(2u))/2.✓ Proved
- \[ = \int \frac{1}{2}\, dx + \int \frac{\cos{\left(6 x + 2 \right)}}{2}\, dx \]linearitySplit the integral into two parts.✓ Proved
- \[ = \frac{x}{2} + \int \frac{\cos{\left(6 x + 2 \right)}}{2}\, dx \]antiderivativeIntegrate the first constant term.✓ Proved
- \[ = \frac{x}{2} + \frac{\sin{\left(6 x + 2 \right)}}{12} \]antiderivative simplifyIntegrate the cosine term using the substitution rule for linear arguments. Simplify the coefficients.✓ Proved
Answer \( \frac{x}{2} + \frac{\sin{\left(6 x + 2 \right)}}{12} + C \)
✓ 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.
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: pass — The solution correctly applies the power-reduction identity, splits the integral using linearity, and computes the antiderivatives step-by-step. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the power-reduction identity, splits the integral using linearity, and computes the antiderivatives step-by-step. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the power-reduction identity, linearity of integration, and standard antiderivative rules. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.