Integral of \( \displaystyle \tan{\left(3 x \right)} \)
Problem 4.1310 · easy Video
Find \( \displaystyle \int \tan{\left(3 x \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \tan{\left(3 x \right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \frac{\sin{\left(3 x \right)}}{\cos{\left(3 x \right)}}\, dx \]trig-identity algebraUse the identity tan(u) = sin(u)/cos(u). Rewrite the integrand to reveal the chain rule pattern.✓ Proved
- \[ = - \frac{\ln{\left(\cos{\left(3 x \right)} \right)}}{3} \]antiderivativeIntegrate using the substitution pattern for the chain rule.✓ Proved
Answer \( - \frac{\ln{\left(\cos{\left(3 x \right)} \right)}}{3} + C \)
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. 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 tan has poles at odd multiples of pi/2 undefined where cos(3*x) = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(3*x) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(3*x) = 0 log is undefined for non-positive arguments |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 log is undefined for non-positive arguments |
| 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 definition of tangent, rearranges the integrand to match the derivative of the logarithm of cosine, and integrates. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-22 — The solution correctly applies the definition of tangent, rearranges the integrand to match the derivative of the logarithm of cosine, and integrates. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.gpt-oss:20b: pass 2026-10-22qwen3.6:27b-mlx: fail (error) 2026-10-21 — Step 3 applies algebraic manipulation to rewrite the integrand, but Step 4 performs the integration (antiderivative) in a single jump without an explicit substitution step. The contract requires each step to change only one thing; combining the algebraic setup and the integration into one step violates the granularity rule. Additionally, Step 4's label 'antiderivative' is vague compared to 'substitution' or 'parts', but the primary defect is the multi-rule application across steps 3 and 4.gpt-oss:20b: pass 2026-10-21qwen3.6:27b-mlx: fail (style) 2026-10-21 — Step 3 applies multiple algebraic manipulations (rewriting tan as sin/cos, factoring out constants, and rearranging terms) simultaneously, violating the one-change-per-step rule. Step 4 jumps from the integral to the final answer without showing the substitution step, making the 'antiderivative' label ambiguous as it skips the explicit u-substitution logic.gpt-oss:20b: pass 2026-10-21
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 2026-10-22 with SymPy 1.14.0.