Question

using the following table of values, find $\int_{0}^{1} f(x) d x$, given $f(x)$ is a continuous function and $f^{prime}(x)=f(x)$. table of values $\

$$\begin{array}{|c|c|c|}\\hline x & f(x) & f(x) \\\\ \\hline 0 & -3 & 5 \\\\ \\hline 1 & 2 & -1 \\\\ \\hline \\end{array}$$

$ -6 6 5 no correct answer choice is given. -5

Explanation:

Step1: Apply the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that if \(F^{\prime}(x)=f(x)\), then \(\int_{a}^{b}f(x)dx = F(b)-F(a)\). Here, \(a = 0\) and \(b = 1\).

Step2: Substitute the values of \(F(x)\) from the table

We know that \(F(0)=5\) and \(F(1)=-1\). So, \(\int_{0}^{1}f(x)dx=F(1)-F(0)\).
Substitute the values: \(\int_{0}^{1}f(x)dx=-1 - 5\).

Answer:

\(-6\)