Integral of \( \displaystyle \frac{\sin{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}} \)
Problem 4.645 · medium
Find \( \displaystyle \int \frac{\sin{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{\sin{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \tan{\left(2 x \right)} \sec{\left(2 x \right)}\, dx \]trig-identityUse the identity sin(u)/cos(u)^2 = tan(u)*sec(u).✓ Proved
- \[ = \frac{0.5}{\cos{\left(2 x \right)}} \]antiderivativeThe antiderivative of sec(2*x)*tan(2*x) is 1/2 * sec(2*x).✓ Proved
Answer \( \frac{1}{2 \cos{\left(2 x \right)}} + C \)
✓ 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 undefined where cos(2*x) = 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 undefined where cos(2*x) = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 undefined where cos(2*x) = 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 trigonometric identities to rewrite the integrand and then uses the standard antiderivative formula for secant times tangent, accounting for the chain rule factor. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly applies trigonometric identities to rewrite the integrand and then uses the standard antiderivative formula for secant times tangent, accounting for the chain rule factor. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly applies trigonometric identities and standard antiderivative rules. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-08
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 2026-10-08 with SymPy 1.14.0.