Substitution with a given u
Problem 4.487 · medium
Evaluate \( \displaystyle \int 2 \tan^{3}{\left(x \right)} \sec^{2}{\left(x \right)}\, dx \) using the substitution \( \displaystyle u = \tan{\left(x \right)} \).
- \[ \frac{d}{d x} \tan{\left(x \right)} = \tan^{2}{\left(x \right)} + 1 \]So du = 1sec²(1x) dx.✓ Proved
- In terms of u the integral becomes ∫ 2*u**3 du.Reviewed
- \[ \frac{d}{d u} \frac{u^{4}}{2} = 2 u^{3} \]An antiderivative in u is u**4/2; differentiating confirms it.✓ Proved
- \[ \frac{\tan^{4}{\left(x \right)}}{2} \]Replace u by tan(x).✓ Proved
- \[ \frac{d}{d x} \frac{\tan^{4}{\left(x \right)}}{2} = 2 \tan^{3}{\left(x \right)} \sec^{2}{\left(x \right)} \]Differentiating the answer returns the integrand.✓ Proved
Answer \( \frac{\tan^{4}{\left(x \right)}}{2} + C \)
✓ 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 and verifies the result. The notation in step 1 is slightly informal but mathematically sound.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the substitution method and verifies the result. The notation in step 1 is slightly informal but mathematically sound.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the substitution method, handles the differential transformation accurately, and verifies the result. The logic is sound and the steps are clear.
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 2026-10-06 with SymPy 1.14.0.