Question

Use the Second Fundamental Theorem of Calculus to find \(F^{\prime}(x)\). $$ F(x)=\int_{0}^{x} t \cos t d t $$

Short answer

The derivative \(F^{\prime}(x)\) of the given function \(F(x)=\int_{0}^{x} t \cos t d t \) is \(x\cos(x)\).

Step-by-step solution

Set Up the Function

The given function is \(F(x)=\int_{0}^{x} t \cos t d t \), where the integrand function \(f(t)\) is \(t\cos(t)\). The derivative of this function would be \(F^{\prime}(x)\ = \ f(x)\).

Use the Second Fundamental Theorem of Calculus

According to the Second Fundamental Theorem of Calculus, if a function \(F\) is defined by an integral with an upper limit of \(x\) and a lower limit a constant, then the derivative of \(F\) is the function we integrated over. In other words, if \(F(x)=\int_{a}^{x} f(t) d t\), then \(F^{\prime}(x) = f(x)\).

Find the Derivative

Applying the Second Fundamental Theorem of Calculus, the derivative of function \(F(x)\) is \(F^{\prime}(x) = f(x) = x\cos(x)\).