Home›Calculus 2›Trigonometric integrals›Problem 4.932

Integral of \( \displaystyle \cos^{3}{\left(x - 1 \right)} \)

Problem 4.932 · easy

Find \( \displaystyle \int \cos^{3}{\left(x - 1 \right)} \, dx \). (Omit the constant of integration.)

\[ \int \cos^{3}{\left(x - 1 \right)}\, dx \]
integral algebraStart with the integral of the given function. Expand the power.✓ Proved
\[ = \int \left(1 - \sin^{2}{\left(x - 1 \right)}\right) \cos{\left(x - 1 \right)}\, dx \]
trig-identityUse the identity cos(x - 1)**2 = 1 - sin(x - 1)**2.✓ Proved
\[ = - \int \sin^{2}{\left(x - 1 \right)} \cos{\left(x - 1 \right)}\, dx + \int \cos{\left(x - 1 \right)}\, dx \]
linearityDistribute the cosine term.✓ Proved
\[ = \sin{\left(x - 1 \right)} - \int \sin^{2}{\left(x - 1 \right)} \cos{\left(x - 1 \right)}\, dx \]
antiderivativeIntegrate the first term.✓ Proved
\[ = \sin{\left(x - 1 \right)} - \int d{\left(\sin{\left(x - 1 \right)} \right)} \sin^{2}{\left(x - 1 \right)}\, dx \]
rewriteRecognize the derivative of sin(x - 1) is cos(x - 1).Not checked
\[ = \sin{\left(x - 1 \right)} - \int \sin^{2}{\left(x - 1 \right)} \cos{\left(x - 1 \right)}\, dx \]
simplifyReturn to the standard form.Not checked
\[ = - \frac{\sin^{3}{\left(x - 1 \right)}}{3} + \sin{\left(x - 1 \right)} \]
antiderivativeIntegrate using the power rule for sin(x - 1).✓ Proved

Answer \( \frac{\left(\cos{\left(2 x - 2 \right)} + 5\right) \sin{\left(x - 1 \right)}}{6} + C \)

Lines: 7 proved, 2 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	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	Not checked	—	simplify left -sin(x - 1)*3/3 + Integral(d(sin(x - 1))sin(x - 1)**2, x); no point in the sample was defined on both lines
7	Not checked	—	simplify left sin(x - 1)*3/3 - Integral(d(sin(x - 1))sin(x - 1)**2, x); no point in the sample was defined on both lines
8	✓ 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: pass
qwen3.6:27b-mlx: fail (error) — Step 6 introduces 'd(sin(x - 1))' inside an integral with respect to x, which is notationally invalid and mathematically incorrect without a corresponding substitution step (dx = d(sin(x-1))/cos(x-1)). Step 7 then reverts to the previous form, making the sequence logically incoherent.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-12
qwen3.6:27b-mlx: fail (error) 2026-10-12 — Step 6 introduces 'd(sin(x - 1))' inside an integral with respect to x, which is notationally invalid and mathematically incorrect without a corresponding substitution step (dx = d(sin(x-1))/cos(x-1)). Step 7 then reverts to the previous form, making the sequence logically incoherent.
gpt-oss:20b: pass 2026-10-12
qwen3.6:27b-mlx: fail (error) 2026-10-12 — Step 6 is a no-op that rewrites the integrand back to its previous form in Step 7, creating a logical loop where the integral is never actually solved via substitution. The transition from Step 5 to Step 8 requires a substitution step (u = sin(x-1)) which is missing or malformed.

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 2026-10-12 with SymPy 1.14.0.