x2sinx

x2sinx refers to the real-valued function f defined by f(x) = x^2 sin x for real x. It is defined on the entire real line, continuous, and differentiable to all orders.

Basic properties include that f is an odd function since f(-x) = -f(x). It has zeros at x = kπ for all integers k, because sin(kπ) = 0. Near the origin, sin x ≈ x, so f(x) ≈ x^3, indicating a small, cubic rise around x = 0.

Derivatives: The first derivative is f'(x) = 2x sin x + x^2 cos x. Higher derivatives exist since the function is smooth, and the growth of the function is driven by the x^2 factor multiplying the bounded sine and cosine terms.

Graph and behavior: As |x| grows, the oscillations of sin x are modulated by the increasingly large envelope x^2, so the amplitude of f(x) increases roughly like x^2. The function continues to oscillate with period 2π and crosses zero at multiples of π.

Antiderivative: An antiderivative is F(x) = ∫ x^2 sin x dx = -x^2 cos x + 2x sin x + 2 cos x + C, obtained by successive integration by parts.

Applications and notes: The function x^2 sin x appears in calculus problems, particularly in integration techniques and in illustrating products of polynomials with trigonometric functions. It can be viewed as the imaginary part of x^2 e^{ix}, linking it to complex exponential methods.