Integral of \( \displaystyle \cos^{3}{\left(3 x + 1 \right)} \)
Problem 4.480 · medium
Find \( \displaystyle \int \cos^{3}{\left(3 x + 1 \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \cos^{3}{\left(3 x + 1 \right)}\, dx \]integral algebraStart with the integral of the given function. Rewrite the power as a product.✓ Proved
- \[ = \int \left(\frac{\cos{\left(6 x + 2 \right)}}{2} + \frac{1}{2}\right) \cos{\left(3 x + 1 \right)}\, dx \]trig-identityUse the identity cos(2u) = 2cos(2) - 1.≈ Checked numerically
- \[ = \int \left(\frac{\cos{\left(3 x + 1 \right)} \cos{\left(6 x + 2 \right)}}{2} + \frac{\cos{\left(3 x + 1 \right)}}{2}\right)\, dx \]linearityDistribute the cosine term.✓ Proved
- \[ = \int \left(\frac{3 \cos{\left(3 x + 1 \right)}}{4} + \frac{\cos{\left(9 x + 3 \right)}}{4}\right)\, dx \]trig-identity algebra algebra algebraUse the product-to-sum identity cos(A)cos(B) = 1/2(cos(A+B) + cos(A-B)). Expand the arguments of the cosines. Simplify the arguments. Combine like terms.≈ Checked numerically
- \[ = \frac{\sin{\left(3 x + 1 \right)}}{4} + \frac{\sin{\left(9 x + 3 \right)}}{36} \]antiderivativeIntegrate each term using the rule for cos(ax+b).✓ Proved
Answer \( \frac{\left(\cos{\left(6 x + 2 \right)} + 5\right) \sin{\left(3 x + 1 \right)}}{18} + C \)
Lines: 8 proved, 2 checked numerically. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 3 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left 2*sin(3*x + 1)**3/9 + sin(3*x + 1)*cos(3*x + 1)**2/3 + sin(3*x + 1)*cos(6*x + 2)/18 - sin(3*x + 1)/6 - sin(6*x + 2)*cos(3*x + 1)/9; numeric agreement only, at 24 of 24 sampled points |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 5 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left -sin(3*x + 1)*cos(6*x + 2)/18 - sin(3*x + 1)/12 + sin(6*x + 2)*cos(3*x + 1)/9 - sin(9*x + 3)/36; numeric agreement only, at 24 of 24 sampled points |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated the stated antiderivative back to the integrand |
Reviewers
gpt-oss:20b: fail (error) — The final antiderivative does not match the correct result obtained by integrating term‑by‑term. The algebraic manipulation leading to the given answer is incorrect.qwen3.6:27b-mlx: fail (error) — Step 3 contains a false mathematical statement in the note ('cos(2u) = 2cos(2) - 1') and applies an incorrect transformation (it should be cos^2(u) = (1+cos(2u))/2, but the step writes 1/2*(1+cos(2*(3x+1))) which is correct for cos^2, yet the note is nonsensical and the label 'trig-identity' is applied to a step that also implicitly handles the power reduction, but the main issue is the note teaches a falsehood). Furthermore, Step 9 applies the antiderivative rule to both terms simultaneously, violating the 'one thing per step' constraint (it should split into two steps or use linearity first).
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-06 — The final antiderivative does not match the correct result obtained by integrating term‑by‑term. The algebraic manipulation leading to the given answer is incorrect.qwen3.6:27b-mlx: fail (error) 2026-10-06 — Step 3 contains a false mathematical statement in the note ('cos(2u) = 2cos(2) - 1') and applies an incorrect transformation (it should be cos^2(u) = (1+cos(2u))/2, but the step writes 1/2*(1+cos(2*(3x+1))) which is correct for cos^2, yet the note is nonsensical and the label 'trig-identity' is applied to a step that also implicitly handles the power reduction, but the main issue is the note teaches a falsehood). Furthermore, Step 9 applies the antiderivative rule to both terms simultaneously, violating the 'one thing per step' constraint (it should split into two steps or use linearity first).gpt-oss:20b: fail (error) 2026-10-06 — The final antiderivative 1/4 sin(3x+1)+1/36 sin(9x+3) is not equivalent to the stated answer (cos(6x+2)+5) sin(3x+1)/18. The solution stops at an incorrect form and does not perform the necessary algebraic simplification to match the given result.qwen3.6:27b-mlx: fail (error) 2026-10-06 — Step 3 cites the identity cos(2u) = 2cos(2) - 1, which is mathematically incorrect (it should be 2cos^2(u) - 1). Although the resulting expression in Step 3 is correct, the stated rule is false and would teach a student incorrect mathematics.
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
gemma4:26b, checked 2026-10-06 with SymPy 1.14.0.