Home›Calculus 1›u-substitution›Problem 4.1497

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

Problem 4.1497 · medium Video

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

\[ \int \sin{\left(x - 1 \right)} \cos^{3}{\left(x - 1 \right)}\, dx \]
integral rewriteStart with the integral of the given function. Rewrite the integrand to show the derivative of cos(x - 1).✓ Proved
\[ = - \frac{\cos^{4}{\left(x - 1 \right)}}{4} \]
antiderivativeIntegrate using the power rule for the substitution u = cos(x - 1).✓ Proved

Answer \( - \frac{\cos^{4}{\left(x - 1 \right)}}{4} + C \)

Top: the graph of F = -cos(x - 1)⁴/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = sin(x - 1)·cos(x - 1)³.

✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation. 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
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 (style) — Step 2 applies two algebraic manipulations (reordering factors and inserting -1 * -1) simultaneously, violating the one-change-per-step rule. Step 3 skips the explicit substitution step, jumping directly to the result.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (style) 2026-11-13 — Step 2 applies two algebraic manipulations (reordering factors and inserting -1 * -1) simultaneously, violating the one-change-per-step rule. Step 3 skips the explicit substitution step, jumping directly to the result.
gpt-oss:20b: pass 2026-11-13
qwen3.6:27b-mlx: pass 2026-11-13 — The solution correctly identifies the substitution structure and applies the power rule for integration. The steps are logically sound and the labels are appropriate.
gpt-oss:20b: pass 2026-11-13

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 2026-11-13 with SymPy 1.14.0.