Question

question if (f(x)=int_{x}^{3}4t^{2}dt), find (f(x)).

Explanation:

Step1: Apply the fundamental theorem of calculus

If \(f(x)=\int_{a}^{x}g(t)dt\), then \(f^{\prime}(x) = g(x)\). Here \(f(x)=\int_{x}^{3}4t^{2}dt=-\int_{3}^{x}4t^{2}dt\).

Step2: Differentiate the integral

By the fundamental - theorem of calculus, if \(F(x)=-\int_{3}^{x}4t^{2}dt\), then \(f^{\prime}(x)=-4x^{2}\).

Answer:

\(-4x^{2}\)