Integrals giving inverse trig functions
Problem 4.641 · medium
Evaluate \( \displaystyle \int \frac{4}{\sqrt{- x^{2} + 4 x - 3}}\, dx \).
- \[ - x^{2} + 4 x - 3 = 1 - \left(x - 2\right)^{2} \]Complete the square.✓ Proved
- With u = x − 2 and a = 1, this is c·∫ du/√(a² − u²)Reviewed
- \[ \frac{d}{d x} 4 \operatorname{asin}{\left(x - 2 \right)} = \frac{4}{\sqrt{- x^{2} + 4 x - 3}} \]An antiderivative is 4*asin(x - 2); differentiate to confirm.✓ Proved
Answer \( 4 \operatorname{asin}{\left(x - 2 \right)} + C \)
✓ 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 = 1.1 |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly completes the square, identifies the standard integral form, and verifies the result via differentiation.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly completes the square, identifies the standard integral form, and verifies the result via differentiation.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly completes the square, identifies the standard integral form for arcsine, and verifies the result by differentiation.
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 2026-10-07 with SymPy 1.14.0.