Substitution with a given u
Problem 4.934 · medium
Evaluate \( \displaystyle \int 4 \left(e^{x} + 1\right)^{3} e^{x}\, dx \) using the substitution \( \displaystyle u = e^{x} + 1 \).
- \[ \frac{d}{d x} \left(e^{x} + 1\right) = e^{x} \]So du = 1e^(1x) dx.✓ Proved
- In terms of u the integral becomes ∫ 4*u**3 du.Reviewed
- \[ \frac{d}{d u} u^{4} = 4 u^{3} \]An antiderivative in u is u**4; differentiating confirms it.✓ Proved
- \[ \left(e^{x} + 1\right)^{4} \]Replace u by exp(x) + 1.✓ Proved
- \[ \frac{d}{d x} \left(e^{x} + 1\right)^{4} = 4 \left(e^{x} + 1\right)^{3} e^{x} \]Differentiating the answer returns the integrand.✓ Proved
Answer \( \left(e^{x} + 1\right)^{4} + C \)
✓ 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: passqwen3.6:27b-mlx: pass — The solution correctly applies the substitution method, handles the differential transformation accurately, and includes the constant of integration in the final answer.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly applies the substitution method, handles the differential transformation accurately, and includes the constant of integration in the final answer.gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: pass 2026-10-12
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 2026-10-12 with SymPy 1.14.0.