Substitution with a given u
Problem 4.937 · hard
Evaluate \( \displaystyle \int 4 \tan^{4}{\left(4 x \right)} \sec^{2}{\left(4 x \right)}\, dx \) using the substitution \( \displaystyle u = \tan{\left(4 x \right)} \).
- \[ \frac{d}{d x} \tan{\left(4 x \right)} = 4 \tan^{2}{\left(4 x \right)} + 4 \]So du = 4sec²(4x) dx.✓ Proved
- In terms of u the integral becomes ∫ u**4 du.
- \[ \frac{d}{d u} \frac{u^{5}}{5} = u^{4} \]An antiderivative in u is u**5/5; differentiating confirms it.✓ Proved
- \[ \frac{\tan^{5}{\left(4 x \right)}}{5} \]Replace u by tan(4*x).✓ Proved
- \[ \frac{d}{d x} \frac{\tan^{5}{\left(4 x \right)}}{5} = 4 \tan^{4}{\left(4 x \right)} \sec^{2}{\left(4 x \right)} \]Differentiating the answer returns the integrand.✓ Proved
Answer \( \frac{\tan^{5}{\left(4 x \right)}}{5} + C \)
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: inconclusive — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"Step 1 incorrectly states the derivative of \(\tan(4x)\) as \(4\tan^2(4x)+4\); the correct derivative is \(4\sec^2(4x)\). This mis‑identifies \(du\) and iqwen3.6:27b-mlx: pass — The solution correctly applies the substitution method and verifies the result by differentiation. The intermediate algebraic steps are sound.
Every verdict on record (4)
gpt-oss:20b: inconclusive 2026-10-12 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"Step 1 incorrectly states the derivative of \(\tan(4x)\) as \(4\tan^2(4x)+4\); the correct derivative is \(4\sec^2(4x)\). This mis‑identifies \(du\) and iqwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly applies the substitution method and verifies the result by differentiation. The intermediate algebraic steps are sound.gpt-oss:20b: fail (error) 2026-10-12 — Step 1 incorrectly states the derivative of tan(4x) as 4·tan²(4x)+4; the correct derivative is 4·sec²(4x). This mis‑statement undermines the substitution justification.qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly applies the substitution method and verifies the result via differentiation. The intermediate equation in step 1 is syntactically incorrect but explicitly labeled as unchecked, so it does not affect the validity of the logical steps.
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.