Integral of \( \displaystyle \cos^{3}{\left(3 x - 1 \right)} \)
Problem 4.1356 · medium Video
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. Expand the power.✓ Proved
- \[ = \int \left(1 - \sin^{2}{\left(3 x - 1 \right)}\right) \cos{\left(3 x - 1 \right)}\, dx \]trig-identityUse the identity cos(u)^2 = 1 - sin(u)^2.✓ Proved
- \[ = - \int \sin^{2}{\left(3 x - 1 \right)} \cos{\left(3 x - 1 \right)}\, dx + \int \cos{\left(3 x - 1 \right)}\, dx \]linearity algebra rewrite substitutionDistribute the cosine term and split the integral. Rearrange the second term for clarity. Prepare for substitution by identifying the derivative of the inner function. The substitution u = 3x - 1 is implicitly handled by the structure.✓ Proved
- \[ = \frac{\sin{\left(3 x - 1 \right)}}{3} - \int \sin^{2}{\left(3 x - 1 \right)} \cos{\left(3 x - 1 \right)}\, dx \]antiderivativeIntegrate the first term.✓ Proved
- \[ = - \frac{\sin^{3}{\left(3 x - 1 \right)}}{9} + \frac{\sin{\left(3 x - 1 \right)}}{3} \]antiderivative simplifyIntegrate the second term using the power rule for sin(u). Simplify the final expression.✓ Proved
Answer \( \frac{\left(\cos{\left(6 x - 2 \right)} + 5\right) \sin{\left(3 x - 1 \right)}}{18} + C \)
Every line of this solution was proved by the computer algebra system SymPy. 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. How we verify
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 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 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 |
| 10 | ✓ 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) — Step 7 claims a substitution has been performed, yet the integrand remains unchanged from step 6. No change in the expression indicates the substitution was not actually applied, making the step invalid.qwen3.6:27b-mlx: fail (error) — Step 7 is labeled 'substitution' but the expression is identical to Step 6 (after simplifying the *3/3 factor), so no substitution rule was applied. Furthermore, Step 9 incorrectly applies the antiderivative rule by introducing an extra factor of 1/3 that does not appear in the integrand of Step 8, resulting in an incorrect coefficient for the final term.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-29 — Step 7 is labeled 'substitution' but the expression is identical to Step 6 (after simplifying the *3/3 factor), so no substitution rule was applied. Furthermore, Step 9 incorrectly applies the antiderivative rule by introducing an extra factor of 1/3 that does not appear in the integrand of Step 8, resulting in an incorrect coefficient for the final term.gpt-oss:20b: fail (error) 2026-10-29 — Step 7 claims a substitution has been performed, yet the integrand remains unchanged from step 6. No change in the expression indicates the substitution was not actually applied, making the step invalid.qwen3.6:27b-mlx: fail (error) 2026-10-28 — Step 6 introduces a factor of 3 and 1/3 (via cos/3 * 3) which is an algebraic manipulation, but Step 7 is labeled 'substitution' and claims the substitution is 'implicitly handled' while the expression remains identical to Step 6 (algebraically). This is a logical non-sequitur: the substitution rule requires changing variables or adjusting the differential, neither of which is explicitly shown or correctly labeled in the transition from 6 to 7. Furthermore, Step 9 applies the antiderivative rule to the integral term, but the coefficient handling is messy and the label 'antiderivative' is applied to a step that effectively performs the integration of the second term, yet the previous steps failed to properly execute the substitution logic required to justify the result. Specifically, Step 7 does not change the expression from Step 6, making the label 'substitution' incorrect as no substitution was performed.gpt-oss:20b: fail (style) 2026-10-28 — Step 7 is labeled "substitution" but no substitution is performed; the integrand is unchanged. The step should be labeled "algebra" or "rewrite". Additionally, step 6 rewrites the integrand by factoring a 3, which is a single algebraic manipulation, but the subsequent labeling of a substitution in step 7 is misleading.qwen3.6:27b-mlx: fail (error) 2026-10-28 — Step 7 is labeled 'substitution' but the expression is identical to Step 6 (after simplifying the *3/3 factor), meaning no substitution was actually performed. Furthermore, Step 9 incorrectly applies the antiderivative rule by introducing a factor of 1/3 that does not appear in the integrand of Step 8, violating the single-rule constraint and producing an algebraically incorrect transition.gpt-oss:20b: fail (error) 2026-10-28 — The solution incorrectly handles the substitution for the &int sin(3x-1)^2 cos(3x-1)dx term. Step 6 rewrites the integrand as sin^2*(cos/3*3) but then step 7 claims a substitution has been applied without actually changing the integral. In step 9 the antiderivative of sin^2(3x-1)cos(3x-1) is taken as (1/3)sin^3(3x-1)/3, which ignores the necessary factor of 3 from the derivative of sin(3x-1). Thus the final expression is mathematically incorrect.
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 2026-10-29 with SymPy 1.14.0.