Integrals giving inverse trig functions
Problem 4.339 · medium
Evaluate \( \displaystyle \int \frac{2}{x^{2} - 6 x + 10}\, dx \).
- \[ x^{2} - 6 x + 10 = \left(x - 3\right)^{2} + 1 \]Complete the square.✓ Proved
- With u = x − 3 and a = 1, this is c·∫ du/(u² + a²)
- \[ \frac{d}{d x} 2 \operatorname{atan}{\left(x - 3 \right)} = \frac{2}{x^{2} - 6 x + 10} \]An antiderivative is 2*atan(x - 3); differentiate to confirm.✓ Proved
Answer \( 2 \operatorname{atan}{\left(x - 3 \right)} + C \)
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 “With u = x − 3 and a = 1, this is c·∫ du/(u² + a²)” omits the constant factor c (which is 2). It also does not explicitly state that the integral becomes 2∫du/(u²+1). This omission could mislead a student into thinking the constant factor is arbitrary or missing.qwen3.6:27b-mlx: pass — The solution correctly completes the square, identifies the standard arctangent integral form, and verifies the result by differentiation. The logic is sound.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly completes the square, identifies the standard arctangent integral form, and verifies the result by differentiation. The logic is sound.gpt-oss:20b: fail (misleading) 2026-10-04 — The sentence “With u = x − 3 and a = 1, this is c·∫ du/(u² + a²)” omits the constant factor c (which is 2). It also does not explicitly state that the integral becomes 2∫du/(u²+1). This omission could mislead a student into thinking the constant factor is arbitrary or missing.qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly completes the square, identifies the standard arctangent integral form, and verifies the result by differentiation. The logic is sound and complete.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.