Final answer:
To find f′(x), the derivative of f(x), we can use the Fundamental Theorem of Calculus. The derivative of the definite integral from a constant to x is equal to f(x). In this case, f′(x) = (1/2)x²cost - (1/2)cost - (1/2)x²sint - (1/2)sint.
Step-by-step explanation:
To find f′(x), the derivative of f(x), we can use the Fundamental Theorem of Calculus. According to the Fundamental Theorem of Calculus, if F(x) is the antiderivative of f(x), then the derivative of the definite integral from a constant to x is equal to f(x). In this case, since f(x) = ∫0x tsint dt, the function F(x) that represents the antiderivative of f(x) is F(x) = (1/2)x²sint - xcost + C. Therefore, f′(x) = (1/2)x²cost - (1/2)cost - (1/2)x²sint - (1/2)sint, where C is the constant of integration.