cos3x2

cos3x2 denotes the real-valued function f(x) = cos(3x^2). This representation uses the cosine of a quadratic argument; to avoid ambiguity, cos(3x^2) is preferred over cos3x2 in formal writing.

The domain of f is all real numbers and the range is [-1, 1]. The function is even, since f(-x) = cos(3(-x)^2) = cos(3x^2) = f(x). The graph oscillates with increasing frequency as |x| grows because the input to cosine, 3x^2, grows quadratically.

Derivative and basic calculus: f'(x) = -6x sin(3x^2). This follows from the chain rule.

Antiderivative: The indefinite integral ∫ cos(3x^2) dx has no elementary closed form. It can be expressed using the Fresnel cosine integral C. In particular, ∫0^x cos(3t^2) dt = sqrt(π/6) · C(x sqrt(6/π)), so an antiderivative is sqrt(π/6) · C(x sqrt(6/π)) + constant.

Series expansion: Around x = 0, cos(3x^2) = 1 - (9/2)x^4 + (81/24)x^8 - …, which gives a convergent Maclaurin series.

Applications and computational aspects: Functions of the form cos(a x^2) arise in optics, diffraction, and signal processing, where quadratic phase terms are common. Because the antiderivative involves Fresnel integrals, numerical evaluation typically relies on special functions or quadrature, using libraries that implement FresnelC.

Notes: Be mindful of notation, as cos(3x^2) differs from (cos(3x))^2 or cos(3x^2) written without parentheses in plain text.