Integrals giving inverse trig functions
Problem 4.505 · medium
Evaluate \( \displaystyle \int \frac{4}{x^{2} + 2 x + 26}\, dx \).
- \[ x^{2} + 2 x + 26 = \left(x + 1\right)^{2} + 25 \]Complete the square.✓ Proved
- With u = x + 1 and a = 5, this is c·∫ du/(u² + a²)
- \[ \frac{d}{d x} \frac{4 \operatorname{atan}{\left(\frac{x}{5} + \frac{1}{5} \right)}}{5} = \frac{4}{x^{2} + 2 x + 26} \]An antiderivative is 4*atan(x/5 + 1/5)/5; differentiate to confirm.✓ Proved
Answer \( \frac{4 \operatorname{atan}{\left(\frac{x}{5} + \frac{1}{5} \right)}}{5} + C \)
Lines: 2 proved, 1 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Not checked | — | 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: fail (misleading) — The sentence claims the integral becomes "c·∫ du/(u² + a²)" but never defines the constant c (it should be 4). This omission misleads a student about the correct scaling factor in the antiderivative.qwen3.6:27b-mlx: pass — The solution correctly completes the square, identifies the standard arctangent integral form, and verifies the result by differentiation.
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-06 — The sentence claims the integral becomes "c·∫ du/(u² + a²)" but never defines the constant c (it should be 4). This omission misleads a student about the correct scaling factor in the antiderivative.qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly completes the square, identifies the standard arctangent integral form, and verifies the result by differentiation.gpt-oss:20b: fail (misleading) 2026-10-06 — The sentence claims the integral becomes "c·∫ du/(u² + a²)" but does not specify that c=4, nor does it show the factor 4 in front of the integral. This omission could mislead a student into thinking the constant factor is irrelevant.qwen3.6:27b-mlx: fail (misleading) 2026-10-06 — The solution omits the constant factor 4 in the transition from the original integral to the standard form, stating 'this is c·∫ du/(u² + a²)' without specifying c=4. This skips the crucial step of factoring out the numerator, which could lead a student to incorrectly apply the formula ∫ du/(u² + a²) directly to the original expression.
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 2026-10-06 with SymPy 1.14.0.