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