Integral of \( \displaystyle \sin{\left(3 x + 1 \right)} + \cos{\left(3 x + 1 \right)} \)
Problem 4.252 · medium
Find \( \displaystyle \int \sin{\left(3 x + 1 \right)} + \cos{\left(3 x + 1 \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \left(\sin{\left(3 x + 1 \right)} + \cos{\left(3 x + 1 \right)}\right)\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \sin{\left(3 x + 1 \right)}\, dx + \int \cos{\left(3 x + 1 \right)}\, dx \]linearitySplit the integral into two parts using linearity.✓ Proved
- \[ = - \frac{\cos{\left(3 x + 1 \right)}}{3} + \int \cos{\left(3 x + 1 \right)}\, dx \]integralIntegrate the sine term.✓ Proved
- \[ = \frac{\sin{\left(3 x + 1 \right)}}{3} - \frac{\cos{\left(3 x + 1 \right)}}{3} \]integralIntegrate the cosine term.✓ Proved
Answer \( \frac{\sin{\left(3 x + 1 \right)} - \cos{\left(3 x + 1 \right)}}{3} + C \)
✓ 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.
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: pass — The solution correctly applies linearity to split the integral and then integrates each term using the standard antiderivative rules for sine and cosine with a linear argument. Each step changes only one thing and uses valid labels.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies linearity to split the integral and then integrates each term using the standard antiderivative rules for sine and cosine with a linear argument. Each step changes only one thing and uses valid labels.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies linearity to split the integral and then integrates each term using standard antiderivative rules with appropriate chain rule factors. All steps change only one thing and use valid labels.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.