Substitution with a given u
Problem 4.780 · hard
Evaluate \( \displaystyle \int \tan^{5}{\left(3 x \right)} \sec^{2}{\left(3 x \right)}\, dx \) using the substitution \( \displaystyle u = \tan{\left(3 x \right)} \).
- \[ \frac{d}{d x} \tan{\left(3 x \right)} = 3 \tan^{2}{\left(3 x \right)} + 3 \]So du = 3sec²(3x) dx.✓ Proved
- In terms of u the integral becomes ∫ u**5/3 du.Reviewed
- \[ \frac{d}{d u} \frac{u^{6}}{18} = \frac{u^{5}}{3} \]An antiderivative in u is u**6/18; differentiating confirms it.✓ Proved
- \[ \frac{\tan^{6}{\left(3 x \right)}}{18} \]Replace u by tan(3*x).✓ Proved
- \[ \frac{d}{d x} \frac{\tan^{6}{\left(3 x \right)}}{18} = \tan^{5}{\left(3 x \right)} \sec^{2}{\left(3 x \right)} \]Differentiating the answer returns the integrand.✓ Proved
Answer \( \frac{\tan^{6}{\left(3 x \right)}}{18} + C \)
Lines: 4 proved, 1 reviewed. 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 | 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 u = tan(3x), handles the differential factor 1/3 properly, and verifies the result by differentiation.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies the substitution u = tan(3x), handles the differential factor 1/3 properly, and verifies the result by differentiation.gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: fail (error) 2026-10-09 — The substitution logic is flawed: if u = tan(3x), then du = 3 sec^2(3x) dx, so sec^2(3x) dx = du/3. The integral becomes ∫ u^5 (du/3) = (1/3) ∫ u^5 du. The solution claims the integral becomes ∫ u^5/3 du, which is algebraically correct, but the derivation in step 1 is garbled (Derivative(tan(3x)) is 3sec^2(3x), not 3tan^2+3). More critically, step 3 claims the antiderivative is u^6/18. Integrating (1/3)u^5 gives (1/3)(u^6/6) = u^6/18. This part is actually correct. However, step 1's equation `Derivative(tan(3*x), x) = 3*tan(3*x)**2 + 3` is mathematically false (it should be 3*sec^2(3x)). While the text comment says `du = 3sec²(3x) dx`, the labeled equation is wrong. A student might learn that the derivative of tan is tan^2+1 (which is true for sec^2, but the coefficient 3 is missing in the tan^2 term if they misread, or they might just be confused by the wrong equation). Actually, d/dx(tan(3x)) = 3 sec^2(3x). The equation shown is 3(tan^2(3x)+1). Since sec^2 = tan^2+1, the equation is technically correct algebraically, just in a different form. Let's re-read carefully. `3*tan(3*x)**2 + 3` is indeed `3*(tan^2(3x)+1) = 3*sec^2(3x)`. So the equation is correct. The substitution leads to `u^5 * (du/3)`. The integral is `1/3 * u^6/6 = u^6/18`. The steps are logically sound and the algebra is correct. Why did I think it was wrong? Step 1 equation is correct. Step 2 sentence is correct. Step 3 equation is correct. Step 4 is correct. Step 5 is correct. Wait, is there a subtle error? The problem asks to evaluate using substitution. The solution does so. The final answer is correct. The intermediate steps are correct. Is there a 'misleading' aspect? No. Is there a 'style' issue? No. It seems correct. Let me double check the derivative in step 1. `Derivative(tan(3*x), x)`. Chain rule: `sec^2(3x) * 3`. Identity: `sec^2 = 1 + tan^2`. So `3(1+tan^2(3x)) = 3 + 3tan^2(3x)`. The equation says `3*tan(3*x)**2 + 3`. This is correct. The comment says `du = 3sec^2(3x) dx`. This is correct. Step 2: `∫ u^5/3 du`. Correct. Step 3: Antiderivative `u^6/18`. Correct. Step 5: Derivative of answer is integrand. Correct. The solution is sound.
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.