Home›Calculus 2›Trigonometric integrals›Problem 4.192

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

Problem 4.192 · medium Video

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

\[ \int \sin{\left(x - 1 \right)} \cos{\left(x - 1 \right)}\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = \int \frac{\sin{\left(2 x - 2 \right)}}{2}\, dx \]
trig-identity algebraUse the double angle identity sin(2u) = 2sin(u)cos(u). Distribute the 2 inside the sine function.✓ Proved
\[ = \frac{\int \sin{\left(2 x - 2 \right)}\, dx}{2} \]
linearityFactor out the constant 1/2.✓ Proved
\[ = - \frac{\cos{\left(2 x - 2 \right)}}{4} \]
antiderivative simplifyIntegrate sin(2x - 2) using the substitution rule for linear arguments. Multiply the constants together.✓ Proved

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

Top: the graph of F = -cos(x - 1)²/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = sin(x - 1)·cos(x - 1).

✓ 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	lines differ by the constant 1/4
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
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: lines differ by the constant 1/4
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: pass — The solution correctly applies the double-angle identity, linearity, and substitution for integration. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies the double-angle identity, linearity, and substitution for integration. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: pass 2026-09-29
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies trigonometric identities, linearity, and substitution to find the antiderivative. The final result is algebraically equivalent to the stated answer.
gpt-oss:20b: pass 2026-09-29

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 2026-10-03 with SymPy 1.14.0.