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Substitution with a given u

Problem 4.622 · medium

Evaluate \( \displaystyle \int \frac{2 x}{x^{2} + 3}\, dx \) using the substitution \( \displaystyle u = x^{2} + 3 \).

\[ \frac{d}{d x} \left(x^{2} + 3\right) = 2 x \]
So du = 2x dx.✓ Proved
In terms of u the integral becomes ∫ 1/u du.
Reviewed
\[ \frac{d}{d u} \ln{\left(u \right)} = \frac{1}{u} \]
An antiderivative in u is log(u); differentiating confirms it.✓ Proved
\[ \ln{\left(x^{2} + 3 \right)} \]
Replace u by x**2 + 3.✓ Proved
\[ \frac{d}{d x} \ln{\left(x^{2} + 3 \right)} = \frac{2 x}{x^{2} + 3} \]
Differentiating the answer returns the integrand.✓ Proved

Answer \( \ln{\left(x^{2} + 3 \right)} + C \)

✓ Nihil obstat Lines: 4 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
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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the substitution method and verifies the result by differentiation. The logic is sound and complete.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly applies the substitution method and verifies the result by differentiation. The logic is sound and complete.
gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: pass 2026-10-07

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 2026-10-07 with SymPy 1.14.0.