Questions: Selected values of the twice-differentiable function g are given in the table above. What is the value of the integral from 0 to 4 of g'(x) * arctan^2(2g(x)+3) dx? x 0 1 2 3 4 g(x) 1 1/8 -3/4 -13/8 -5/2

Selected values of the twice-differentiable function g are given in the table above. What is the value of the integral from 0 to 4 of g'(x) * arctan^2(2g(x)+3) dx?

x 0 1 2 3 4
g(x) 1 1/8 -3/4 -13/8 -5/2

Transcript text: \begin{tabular}{|c|c|c|c|c|c|} \hline$x$ & 0 & 1 & 2 & 3 & 4 \\ \hline$g(x)$ & 1 & $\frac{1}{8}$ & $-\frac{3}{4}$ & $-\frac{13}{8}$ & $-\frac{5}{2}$ \\ \hline \end{tabular} Selected values of the twice-differentiable function $g$ are given in the table above. What is the value of $\int_{0}^{4} g^{\prime}(x) \arctan ^{2}(2 g(x)+3) d x ?$

Solution

Solution Steps

Step 1: Define the Function $ g(x) $

The function $ g(x) $ is defined based on the values provided in the table:

\[ g(0) = 1, \quad g(1) = \frac{1}{8}, \quad g(2) = -\frac{3}{4}, \quad g(3) = -\frac{13}{8}, \quad g(4) = -\frac{5}{2} \]

Step 2: Set Up the Integral

We need to evaluate the integral

\[ \int_{0}^{4} g^{\prime}(x) \arctan^{2}(2 g(x) + 3) \, dx \]

According to the Fundamental Theorem of Calculus, this integral can be simplified to:

\[ \arctan^{2}(2 g(4) + 3) - \arctan^{2}(2 g(0) + 3) \]

Step 3: Evaluate the Endpoints

Now we calculate the values at the endpoints $ x = 0 $ and $ x = 4 $:

For $ x = 0 $: \[ 2 g(0) + 3 = 2 \cdot 1 + 3 = 5 \quad \Rightarrow \quad \arctan^{2}(5) \]
For $ x = 4 $: \[ 2 g(4) + 3 = 2 \cdot \left(-\frac{5}{2}\right) + 3 = -5 + 3 = -2 \quad \Rightarrow \quad \arctan^{2}(-2) \]

Step 4: Calculate the Final Result

The final result of the integral is:

\[ \arctan^{2}(-2) - \arctan^{2}(5) \]

Evaluating this expression gives us the numerical result:

\[ \text{result} = -0.6604513833320595 \]

This value represents the outcome of the integral $ \int_{0}^{4} g^{\prime}(x) \arctan^{2}(2 g(x) + 3) \, dx $.