find the most general antiderivative of the function. (check your answer by differentiation. use c for the…

Explanation:

Step1: Simplify the function

First, rewrite $f(t)$ as $f(t)=\frac{2t}{\sqrt{t}}-\frac{4}{\sqrt{t}}+\frac{4\sqrt{t}}{\sqrt{t}} = 2t^{\frac{1}{2}}-4t^{-\frac{1}{2}} + 4$.

Step2: Integrate term - by - term

Using the power rule for integration $\int t^n dt=\frac{t^{n + 1}}{n+1}+C$ ($n\neq - 1$), we have: For the first term $\int 2t^{\frac{1}{2}}dt=2\times\frac{t^{\frac{1}{2}+1}}{\frac{1}{2}+1}=\frac{4}{3}t^{\frac{3}{2}}$. For the second term $\int - 4t^{-\frac{1}{2}}dt=-4\times\frac{t^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}=-8t^{\frac{1}{2}}$. For the third term $\int 4dt = 4t$.

Step3: Combine the results

The antiderivative $F(t)=\frac{4}{3}t^{\frac{3}{2}}-8t^{\frac{1}{2}}+4t + C$.

Answer:

$F(t)=\frac{4}{3}t^{\frac{3}{2}}-8t^{\frac{1}{2}}+4t + C$