Question

Consider the function whose formula is given by f(x) = x sin(2x).

a) f(x) = sin(2x) + x cos(2x)
b) f(x) = cos(2x) + x sin(2x)
c) f(x) = 2x cos(2x)
d) f(x) = 2 cos(2x) + x sin(2x)

Answer 1

The function f(x) = x sin(2x) can be expanded using the product-to-sum identities for sine and cosine. The correct expansion is f(x) = x (sin(2x)) = x (sin(x) cos(2x) + cos(x) sin(2x)). So, the correct option is b) f(x) = cos(2x) + x sin(2x).

Step-by-step explanation:

The function f(x) = x sin(2x) can be expanded using the product-to-sum identities for sine and cosine. The correct expansion is f(x) = x (sin(2x)) = x (sin(x) cos(2x) + cos(x) sin(2x)). Simplifying further, we get f(x) = x cos(x) sin(2x) + x sin(x) cos(2x).

So, the correct option is b) f(x) = cos(2x) + x sin(2x).