Substitution with a given u
Problem 4.775 · hard
Evaluate \( \displaystyle \int \frac{6 x}{x^{2} + 4}\, dx \) using the substitution \( \displaystyle u = x^{2} + 4 \).
- \[ \frac{d}{d x} \left(x^{2} + 4\right) = 2 x \]So du = 2x dx.✓ Proved
- In terms of u the integral becomes ∫ 3/u du.
- \[ \frac{d}{d u} 3 \ln{\left(u \right)} = \frac{3}{u} \]An antiderivative in u is 3*log(u); differentiating confirms it.✓ Proved
- \[ 3 \ln{\left(x^{2} + 4 \right)} \]Replace u by x**2 + 4.✓ Proved
- \[ \frac{d}{d x} 3 \ln{\left(x^{2} + 4 \right)} = \frac{6 x}{x^{2} + 4} \]Differentiating the answer returns the integrand.✓ Proved
Answer \( 3 \ln{\left(x^{2} + 4 \right)} + C \)
Lines: 4 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 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | ✓ 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 | a difference quotient of the answer matches the integrand at x = 0.4, 0.9, 1.3 |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (misleading) — The solution omits the absolute value in the logarithm, writing 3 ln(x^2 + 4) instead of 3 ln|x^2 + 4|. While x^2 + 4 is always positive, omitting the absolute value in the general formula for ∫(1/u)du teaches an incorrect general rule.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: fail (misleading) 2026-10-09 — The solution omits the absolute value in the logarithm, writing 3 ln(x^2 + 4) instead of 3 ln|x^2 + 4|. While x^2 + 4 is always positive, omitting the absolute value in the general formula for ∫(1/u)du teaches an incorrect general rule.gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: fail (misleading) 2026-10-09 — The solution omits the constant of integration (+ C) in the intermediate steps and the final stated answer, which is a critical component of indefinite integrals.
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/indefinite_substitution, checked 2026-10-09 with SymPy 1.14.0.