Trigonometric substitution
Problem 4.576 · hard
Evaluate \( \displaystyle \int \frac{1}{\sqrt{x^{2} + 4}}\, dx \).
- The radical suggests x = 2 tan θ.Reviewed
- \[ 4 \tan^{2}{\left(\theta \right)} + 4 = 4 \sec^{2}{\left(\theta \right)} \]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.Reviewed
- \[ \frac{d}{d x} \ln{\left(x + \sqrt{x^{2} + 4} \right)} = \frac{1}{\sqrt{x^{2} + 4}} \]Differentiating the answer returns the integrand.✓ Proved
Answer \( \ln{\left(x + \sqrt{x^{2} + 4} \right)} + C \)
Lines: 2 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: pass — The solution correctly identifies the trigonometric substitution and verifies the result by differentiation. The steps are logically sound and mathematically accurate.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the trigonometric substitution and verifies the result by differentiation. The steps are logically sound and mathematically accurate.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The solution skips the actual integration steps and the back-substitution process, jumping directly from the substitution setup to the final answer. It fails to demonstrate the integration of sec(theta) or the conversion back to x, making it an incomplete and pedagogically unsound derivation.
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.