Trigonometric substitution
Problem 4.867 · hard
Evaluate \( \displaystyle \int \frac{1}{x^{2} \sqrt{x^{2} + 1}}\, dx \).
- 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.
- \[ \frac{d}{d x} \left(- \frac{\sqrt{x^{2} + 1}}{x}\right) = \frac{1}{x^{2} \sqrt{x^{2} + 1}} \]Differentiating the answer returns the integrand.✓ Proved
Answer \( - \frac{\sqrt{x^{2} + 1}}{x} + C \)
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 (should be x = 1*tan(theta) or just x = tan(theta)), but more critically, the solution skips the actual integration steps and the back-substitution process, merely stating the plan. While the final answer is correct, the solution fails to demonstrate the calculus required to arrive at it, making it an incomplete and misleading guide for a student learning the method.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: fail (error) 2026-10-10 — The substitution x = tan(theta) is missing the coefficient '1' in the text description (should be x = 1*tan(theta) or just x = tan(theta)), but more critically, the solution skips the actual integration steps and the back-substitution process, merely stating the plan. While the final answer is correct, the solution fails to demonstrate the calculus required to arrive at it, making it an incomplete and misleading guide for a student learning the method.gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: fail (error) 2026-10-10 — The substitution x = tan(theta) is missing the coefficient '1' in the text (written as '1 tan theta' instead of 'tan theta'), but more importantly, the solution skips the actual integration steps and verification of the substitution process, jumping straight to a derivative check of the final answer. While the final answer is correct, the 'solution' provided does not actually solve the integral; it only verifies the answer. A solution must show the derivation, not just the check. Furthermore, the substitution x = tan(theta) implies dx = sec^2(theta) dtheta, and sqrt(x^2+1) = sec(theta). The integral becomes int (1/tan^2(theta) * 1/sec(theta)) * sec^2(theta) dtheta = int sec(theta)/tan^2(theta) dtheta = int csc(theta) cot(theta) dtheta = -csc(theta) + C. Converting back: csc(theta) = hyp/opp = sqrt(x^2+1)/x. So -sqrt(x^2+1)/x + C. The provided text fails to show this derivation, making it an incomplete solution rather than a valid one. However, looking at the strict instructions: 'Each line... is either an EQUATION... or a SENTENCE'. Line 1 is a sentence. Line 3 is a sentence. Line 3 says 'Rewrite... integrate... use a right triangle'. It describes the method but does not execute it. Is this an error? Yes, because it claims to be a solution but provides no mathematical steps for the integration itself, only a verification. A student would not learn how to integrate it. It is misleading/incomplete.
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 2026-10-10 with SymPy 1.14.0.