Question

Use the Second Fundamental Theorem of Calculus to find \(F^{\prime}(x)\). $$ F(x)=\int_{0}^{x} \sec ^{3} t d t $$

Short answer

The derivative \(F'(x)\) of the given function \(F(x)\) is \(\sec^{3}(x)\).

Step-by-step solution

Identify the function in the integral

The function in the integral to be used according to Fundamental Theorem of Calculus is \(f(t) = \sec^{3}(t)\).

Apply the Second Fundamental Theorem of Calculus

The Second Fundamental Theorem of Calculus states that the derivative with respect to \(x\) of the integral of \(f(t)\) from a constant to \(x\) is simply \(f(x)\). Therefore, \( F'(x) = f(x) \). Apply this theorem to get \( F'(x) = \sec^{3}(x) \).