Integrals giving inverse trig functions
Problem 4.504 · medium
Evaluate \( \displaystyle \int_{-3}^{-1} \frac{2}{x^{2} + 6 x + 13}\, dx \).
- \[ x^{2} + 6 x + 13 = \left(x + 3\right)^{2} + 4 \]Complete the square.✓ Proved
- With u = x + 3 and a = 2, this is c·∫ du/(u² + a²)
- \[ \frac{d}{d x} \operatorname{atan}{\left(\frac{x}{2} + \frac{3}{2} \right)} = \frac{2}{x^{2} + 6 x + 13} \]An antiderivative is atan(x/2 + 3/2); differentiate to confirm.✓ Proved
- \[ \frac{\pi}{4} \]Evaluate between the limits.✓ Proved
Answer \( \frac{\pi}{4} \)
Lines: 3 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see 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 |
| 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 | numerical quadrature |
Reviewers
gpt-oss:20b: fail (misleading) — The solution’s sentences omit the change of limits and incorrectly introduce an unnecessary constant c. A student would not learn how to apply the substitution to the definite integral or how the limits transform, which is essential for evaluating the integral.qwen3.6:27b-mlx: fail (error) — The antiderivative in step 3 is incorrect; the derivative of atan(x/2 + 3/2) is 2/(x^2 + 6x + 13) only if the chain rule factor 1/2 is accounted for, but the standard integral of 1/(u^2+a^2) is (1/a)atan(u/a). Here a=2, so the antiderivative should be (1/2)atan((x+3)/2). The solution misses the factor 1/2, leading to an incorrect result (the correct answer is pi/8, not pi/4).
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-06 — The solution’s sentences omit the change of limits and incorrectly introduce an unnecessary constant c. A student would not learn how to apply the substitution to the definite integral or how the limits transform, which is essential for evaluating the integral.qwen3.6:27b-mlx: fail (error) 2026-10-06 — The antiderivative in step 3 is incorrect; the derivative of atan(x/2 + 3/2) is 2/(x^2 + 6x + 13) only if the chain rule factor 1/2 is accounted for, but the standard integral of 1/(u^2+a^2) is (1/a)atan(u/a). Here a=2, so the antiderivative should be (1/2)atan((x+3)/2). The solution misses the factor 1/2, leading to an incorrect result (the correct answer is pi/8, not pi/4).gpt-oss:20b: fail (misleading) 2026-10-06 — The solution’s sentences are incomplete and misleading. Sentence 2 incorrectly introduces a constant c and does not state the integral in terms of u. Sentence 4 merely repeats the answer without showing the evaluation of the antiderivative at the limits, so a student would not learn how the result is obtained.qwen3.6:27b-mlx: fail (error) 2026-10-06 — The antiderivative in line 3 is incorrect; it is missing a factor of 1/2. The correct antiderivative of 2/((x+3)^2 + 4) is atan((x+3)/2), not atan(x/2 + 3/2) (which differentiates to 1/((x+3)^2 + 4)). Although the final numerical answer happens to be correct due to a compensating error in the evaluation step, the stated antiderivative is mathematically false.
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.