Home›Calculus 1›Basic antiderivatives›Problem 4.1426

Integral of \( \displaystyle \cos{\left(4 x - 2 \right)} \)

Problem 4.1426 · easy Video

Find \( \displaystyle \int \cos{\left(4 x - 2 \right)} \, dx \). (Omit the constant of integration.)

\[ \int \cos{\left(4 x - 2 \right)}\, dx \]
integralStart with the integral.✓ Proved
\[ = \frac{\sin{\left(4 x - 2 \right)}}{4} \]
antiderivativeIntegrate using the reverse chain rule.✓ Proved

Answer \( \frac{\sin{\left(4 x - 2 \right)}}{4} + C \)

Top: the graph of F = sin(4x - 2)/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = cos(4x - 2).

✓ 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
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: pass — The solution correctly applies the antiderivative rule for a composite trigonometric function, accounting for the chain rule factor. The steps are minimal and correctly labeled.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-11-04 — The solution correctly applies the antiderivative rule for a composite trigonometric function, accounting for the chain rule factor. The steps are minimal and correctly labeled.
gpt-oss:20b: pass 2026-11-04
qwen3.6:27b-mlx: pass 2026-11-03
gpt-oss:20b: pass 2026-11-03
qwen3.6:27b-mlx: pass 2026-11-03
gpt-oss:20b: pass 2026-11-03

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 2026-11-04 with SymPy 1.14.0.