Integrals giving inverse trig functions
Problem 4.340 · medium
Evaluate \( \displaystyle \int \frac{2}{x^{2} + 4 x + 13}\, dx \).
- \[ x^{2} + 4 x + 13 = \left(x + 2\right)^{2} + 9 \]Complete the square.✓ Proved
- With u = x + 2 and a = 3, this is c·∫ du/(u² + a²)Reviewed
- \[ \frac{d}{d x} \frac{2 \operatorname{atan}{\left(\frac{x}{3} + \frac{2}{3} \right)}}{3} = \frac{2}{x^{2} + 4 x + 13} \]An antiderivative is 2*atan(x/3 + 2/3)/3; differentiate to confirm.✓ Proved
Answer \( \frac{2 \operatorname{atan}{\left(\frac{x}{3} + \frac{2}{3} \right)}}{3} + C \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 3 | ✓ 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 | difference quotient of the answer at x = -4.0 |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly completes the square and identifies the standard arctangent integral form. The final verification by differentiation confirms the result is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly completes the square and identifies the standard arctangent integral form. The final verification by differentiation confirms the result is correct.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly completes the square and identifies the standard arctangent integral form. The final verification by differentiation confirms the result is correct.gpt-oss:20b: pass 2026-10-04
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/inverse_trig_integral, checked 2026-10-04 with SymPy 1.14.0.