Integral of \( \displaystyle \sec^{2}{\left(2 x \right)} \)
Problem 4.921 · easy
Find \( \displaystyle \int \sec^{2}{\left(2 x \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \sec^{2}{\left(2 x \right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \frac{1}{\cos^{2}{\left(2 x \right)}}\, dx \]rewriteUse the identity sec(u) = 1/cos(u).✓ Proved
- \[ = \int \sec^{2}{\left(2 x \right)}\, dx \]algebraRewrite the squared term.✓ Proved
- \[ = \int \frac{\sec{\left(2 x \right)}}{\cos{\left(2 x \right)}}\, dx \]algebra algebra algebraExpress one sec(2*x) in terms of cosine. Multiply by 2/2 to prepare for substitution. Simplify the fraction.✓ Proved
- \[ = \int \sec^{2}{\left(2 x \right)}\, dx \]rewrite simplifyConvert 1/cos(2*x) back to sec(2*x). Simplify the integrand.✓ Proved
- \[ = \frac{\tan{\left(2 x \right)}}{2} \]antiderivativeThe antiderivative of sec(2x)^2 is tan(2x)/2.✓ Proved
Answer \( \frac{\tan{\left(2 x \right)}}{2} + C \)
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 sec has poles at odd multiples of pi/2 undefined where cos(2*x) = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(2*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 undefined where cos(2*x) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 sec has poles at odd multiples of pi/2 undefined where cos(2*x) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 sec has poles at odd multiples of pi/2 undefined where cos(2*x) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 sec has poles at odd multiples of pi/2 undefined where cos(2*x) = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 sec has poles at odd multiples of pi/2 |
| 9 | ✓ 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: passqwen3.6:27b-mlx: pass
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: pass 2026-10-12qwen3.6:27b-mlx: fail (style) 2026-10-12 — The solution contains redundant circular steps (steps 2-8) that rewrite the integrand into an equivalent form and then back to the original, adding no value. Step 9 jumps to the final answer without showing the standard substitution or chain rule application required to derive it from the integral form.gpt-oss:20b: fail (error) 2026-10-12 — Step 7 applies two transformations at once: it rewrites 1/cos(2*x) as sec(2*x) and simultaneously simplifies the factor 2*(1/2). According to the contract, each step must change only one thing and use a single rule from the allowed vocabulary.
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 2026-10-12 with SymPy 1.14.0.