Integral of \( \displaystyle \frac{\sin{\left(x + 1 \right)}}{\cos^{2}{\left(x + 1 \right)}} \)
Problem 4.280 · medium
Find \( \displaystyle \int \frac{\sin{\left(x + 1 \right)}}{\cos^{2}{\left(x + 1 \right)}} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{\sin{\left(x + 1 \right)}}{\cos^{2}{\left(x + 1 \right)}}\, dx \]integral algebraStart with the integral of the given function. Rewrite the denominator using a negative exponent.✓ Proved
- \[ = \int \frac{\tan{\left(x + 1 \right)}}{\cos{\left(x + 1 \right)}}\, dx \]trig-identityUse the identity sin(u)/cos(u) = tan(u).Reviewed
- \[ = \int \tan{\left(x + 1 \right)} \sec{\left(x + 1 \right)}\, dx \]trig-identityUse the identity 1/cos(u) = sec(u).Reviewed
- \[ = \sec{\left(x + 1 \right)} \]antiderivativeThe antiderivative of tan(u)sec(u) is sec(u).✓ Proved
Answer \( \frac{1}{\cos{\left(x + 1 \right)}} + C \)
Lines: 4 proved, 2 reviewed. 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(x + 1) = 0 |
| 3 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | simplify left -Integral(tan(x + 1)/cos(x + 1), x) + 1/cos(x + 1); no point in the sample was defined on both lines undefined where cos(x + 1) = 0 tan has poles at odd multiples of pi/2 |
| 4 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | simplify left -sec(x + 1) + Integral(tan(x + 1)/cos(x + 1), x); no point in the sample was defined on both lines tan has poles at odd multiples of pi/2 undefined where cos(x + 1) = 0 sec has poles at odd multiples of pi/2 |
| 5 | ✓ 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(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: passqwen3.6:27b-mlx: pass — The solution correctly applies trigonometric identities to rewrite the integrand as tan(x+1)sec(x+1) and then identifies the antiderivative. Each step applies a single rule from the allowed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies trigonometric identities to rewrite the integrand as tan(x+1)sec(x+1) and then identifies the antiderivative. Each step applies a single rule from the allowed vocabulary.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies algebraic rewriting and trigonometric identities to transform the integrand into a standard form, then applies the known antiderivative rule. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.gpt-oss:20b: fail (error) 2026-10-04 — Step 3 applies two transformations at once: it replaces sin(x+1)/cos(x+1)^2 with tan(x+1)*sec(x+1), using both sin/cos = tan and 1/cos = sec. This violates the rule that each step must change only one thing and be labeled with a single 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 2026-10-04 with SymPy 1.14.0.