Integral of \( \displaystyle \frac{\sin{\left(3 x + 1 \right)}}{\cos^{2}{\left(3 x + 1 \right)}} \)
Problem 4.730 · medium
Find \( \displaystyle \int \frac{\sin{\left(3 x + 1 \right)}}{\cos^{2}{\left(3 x + 1 \right)}} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{\sin{\left(3 x + 1 \right)}}{\cos^{2}{\left(3 x + 1 \right)}}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \tan{\left(3 x + 1 \right)} \sec{\left(3 x + 1 \right)}\, dx \]rewrite simplifyRewrite the integrand using tangent and secant functions. The expression is already in a standard form.✓ Proved
- \[ = \frac{\sec{\left(3 x + 1 \right)}}{3} \]substitutionUse substitution u = 3*x + 1, which implies du = 3dx.✓ Proved
Answer \( \frac{1}{3 \cos{\left(3 x + 1 \right)}} + C \)
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 | simplify(a - b) reduced to 0 undefined where cos(3*x + 1) = 0 tan has poles at odd multiples of pi/2 sec has poles at odd multiples of pi/2 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 sec has poles at odd multiples of pi/2 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 sec has poles at odd multiples of pi/2 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where cos(3*x + 1) = 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated the stated antiderivative back to the integrand |
Reviewers
gpt-oss:20b: fail (error) — Step 4 applies two rules at once: it performs the substitution u=3x+1 and simultaneously uses the antiderivative of sec(u)tan(u). The step should be split into a substitution step followed by an antiderivative step, each labeled appropriately.qwen3.6:27b-mlx: pass — The solution correctly identifies the integrand as a derivative of secant via substitution. The steps are logically sound, labels are appropriate, and the final answer matches the stated result.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-09 — Step 4 applies two rules at once: it performs the substitution u=3x+1 and simultaneously uses the antiderivative of sec(u)tan(u). The step should be split into a substitution step followed by an antiderivative step, each labeled appropriately.qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly identifies the integrand as a derivative of secant via substitution. The steps are logically sound, labels are appropriate, and the final answer matches the stated result.gpt-oss:20b: fail (error) 2026-10-09 — Step 4 applies two rules at once: it performs the substitution u=3x+1 and simultaneously integrates the resulting sec(u)·tan(u) term. The label lists only "substitution", missing the integral rule, violating the one‑rule‑per‑step requirement.qwen3.6:27b-mlx: pass 2026-10-09
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 2026-10-09 with SymPy 1.14.0.