2xsinx

2x sin x denotes the real-valued function f(x) = 2x sin x for all real x. It is the product of the linear factor x and the sine function, scaled by a factor of two. This simple combination is commonly used in calculus and analysis as a representative nontrivial example of a product of elementary functions.

Properties and analysis: The function is even, since f(-x) = f(x). It is continuous and infinitely differentiable on the real line. The zeros occur at x = nπ for integers n (including zero). The first derivative is f'(x) = 2 sin x + 2x cos x, and the second derivative is f''(x) = 4 cos x − 2x sin x. Critical points satisfy 2 sin x + 2x cos x = 0, i.e., tan x = −x, which does not yield a simple closed-form solution. The function is oscillatory due to the sine factor, with its overall magnitude growing roughly linearly with |x| because of the multiplying x.

Antiderivative and related formulas: An antiderivative is ∫ 2x sin x dx = −2x cos x + 2 sin x + C, obtainable by integration by parts. The Maclaurin series for sin x yields 2x sin x = 2x^2 − (2/3!) x^4 + (2/5!) x^6 − ..., showing the function’s even, rapidly increasing powers near x = 0.

Applications and context: 2x sin x appears in various calculus problems, including integration by parts, Fourier analysis, and the construction of test functions. Its simple form makes it useful for illustrating product rules, symmetry properties, and the behavior of oscillatory functions with linearly growing amplitude.