Trigonometric substitution
Problem 4.577 · hard
Evaluate \( \displaystyle \int \frac{1}{\sqrt{x^{2} + 1}}\, dx \).
- The radical suggests x = 1 tan θ.
- \[ \tan^{2}{\left(\theta \right)} + 1 = \sec^{2}{\left(\theta \right)} \]The identity that clears the square root.✓ Proved
- Rewrite the integral in θ, integrate, then use a right triangle to express the trig functions of θ in terms of x.
- \[ \frac{d}{d x} \ln{\left(x + \sqrt{x^{2} + 1} \right)} = \frac{1}{\sqrt{x^{2} + 1}} \]Differentiating the answer returns the integrand.✓ Proved
Answer \( \ln{\left(x + \sqrt{x^{2} + 1} \right)} + C \)
Lines: 2 proved, 2 not checked. 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 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | Not checked | — | a sentence; read, not computed |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | F(b) − F(a) matches numerical quadrature of the integrand on a sub-interval of the domain |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The substitution x = tan(theta) is missing the coefficient '1' in the text description (written as '1 tan theta' instead of 'tan theta' or '1 * tan theta'), but more critically, the solution skips the actual integration steps and the back-substitution process, jumping directly to verifying the answer. While the verification is correct, the solution fails to demonstrate *how* to evaluate the integral as requested, providing only a check rather than a derivation.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The substitution x = tan(theta) is missing the coefficient '1' in the text description (written as '1 tan theta' instead of 'tan theta' or '1 * tan theta'), but more critically, the solution skips the actual integration steps and the back-substitution process, jumping directly to verifying the answer. While the verification is correct, the solution fails to demonstrate *how* to evaluate the integral as requested, providing only a check rather than a derivation.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution claims to use trigonometric substitution (x = tan θ) but fails to actually perform the substitution, integration, or back-substitution steps. Instead, it skips directly to verifying the answer by differentiation, which does not demonstrate the requested method.
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
generator:structured/trig_substitution, checked 2026-10-07 with SymPy 1.14.0.