Integral of \( \displaystyle x \sec^{2}{\left(x \right)} \)
Problem 4.365 · easy
Find \( \displaystyle \int x \sec^{2}{\left(x \right)} \, dx \). (Omit the constant of integration.)
- \[ \int x \sec^{2}{\left(x \right)}\, dx \]integralStart with the integral of the integrand.✓ Proved
- \[ = x \tan{\left(x \right)} - \int \tan{\left(x \right)}\, dx \]partsApply integration by parts with u = x and dv = sec(x)**2 dx.✓ Proved
- \[ = x \tan{\left(x \right)} + \ln{\left(\cos{\left(x \right)} \right)} \]antiderivative algebraThe antiderivative of tan(x) is -log(cos(x)). Simplify the signs.✓ Proved
Answer \( x \tan{\left(x \right)} + \ln{\left(\cos{\left(x \right)} \right)} + C \)
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 tan 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 log is undefined for non-positive arguments |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments 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 log is undefined for non-positive arguments 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 (style) — [domain objection, downgraded to style] The antiderivative of tan(x) is -ln|cos(x)|, not -ln(cos(x)). Omitting the absolute value can mislead students about the domain of the result.qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05gpt-oss:20b: fail (style) 2026-10-05 — [domain objection, downgraded to style] The antiderivative of tan(x) is -ln|cos(x)|, not -ln(cos(x)). Omitting the absolute value can mislead students about the domain of the result.qwen3.6:27b-mlx: pass 2026-10-05gpt-oss:20b: fail (misleading) 2026-10-05 — The antiderivative of tan(x) is -ln|cos(x)|, not -ln(cos(x)). The solution omits the absolute value, which is incorrect for values where cos(x)<0, leading to a misleading result.
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 2026-10-05 with SymPy 1.14.0.