Integral of \( \displaystyle \sec^{2}{\left(3 x \right)} \)
Problem 4.1219 · easy Video
Find \( \displaystyle \int \sec^{2}{\left(3 x \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \sec^{2}{\left(3 x \right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \frac{1}{\cos^{2}{\left(3 x \right)}}\, dx \]rewriteRewrite secant in terms of cosine.✓ Proved
- \[ = \int \sec^{2}{\left(3 x \right)}\, dx \]substitution algebraWe will use the substitution 3*x = v, but we must express everything in terms of x. Prepare for integration by recognizing the derivative of the inner function.✓ Proved
- \[ = \frac{\tan{\left(3 x \right)}}{3} \]antiderivativeThe antiderivative of sec(ax)^2 is (1/a)tan(ax).✓ Proved
Answer \( \frac{\tan{\left(3 x \right)}}{3} + C \)
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. A reviewer's objection to the explanation stands; see 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 sec 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 sec has poles at odd multiples of pi/2 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 sec has poles at odd multiples of pi/2 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 sec has poles at odd multiples of pi/2 tan 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 tan has poles at odd multiples of pi/2 |
| 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 3 incorrectly claims a substitution but does not change the integrand; step 4 applies two operations (introducing a factor of 3 and a factor of 1/3) in one algebraic step, violating the one‑change rule.qwen3.6:27b-mlx: fail (error) — Step 3 is logically incoherent: it claims to use substitution but leaves the expression unchanged, failing to actually perform the substitution. Step 4 introduces a factor of 3 and 1/3 without a valid algebraic justification (it is not an identity, nor a proper substitution step).
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — Step 3 is logically incoherent: it claims to use substitution but leaves the expression unchanged, failing to actually perform the substitution. Step 4 introduces a factor of 3 and 1/3 without a valid algebraic justification (it is not an identity, nor a proper substitution step).gpt-oss:20b: fail (error) 2026-10-18 — Step 3 incorrectly claims a substitution but does not change the integrand; step 4 applies two operations (introducing a factor of 3 and a factor of 1/3) in one algebraic step, violating the one‑change rule.qwen3.6:27b-mlx: fail (error) 2026-10-16 — Step 4 is mathematically invalid; it inserts a factor of 3 and 1/3 without adjusting the differential (dx), which violates the rules of algebra and substitution. Step 5 incorrectly labels the final evaluation as 'antiderivative' when it applies the result of the substitution/integration process, but the primary defect is the nonsensical algebraic manipulation in Step 4.gpt-oss:20b: fail (error) 2026-10-16 — Step 3 incorrectly claims a substitution but leaves the integrand unchanged, and step 4 applies two changes at once (introducing a factor 3 and a factor 1/3) in a single algebraic step, violating the one‑change rule.qwen3.6:27b-mlx: fail (error) 2026-10-15 — Step 4 is mathematically incorrect; it claims that sec(3x)^2 is equal to 3 * sec(3x)^2 * (1/3) inside the integral, which is a tautology presented as a transformation, but the label 'algebra' does not justify the implicit substitution logic required to proceed to Step 5. More critically, Step 5 jumps from the integral to the final answer without showing the substitution variable change or the integration of the constant multiple properly, effectively skipping the actual integration step while mislabeling the previous step. The transition from Step 3 to 4 is also flawed as Step 3 is just a note, not a mathematical transformation, and Step 4 introduces a factor of 3 and 1/3 without a clear rule application that changes the expression's form meaningfully in a single step as required.gpt-oss:20b: fail (error) 2026-10-15 — Step 3 claims to use a substitution but the integrand is unchanged; the substitution rule is not applied. The step incorrectly labels the operation, violating the one‑rule‑per‑step rule.
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 2026-10-18 with SymPy 1.14.0.