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Let
g be a differentiable function such that
g(-2)=-5 and
g^(')(x)=(x^(2))/(cos^(2)(x)+1).
What is the value of
g(4) ?
Use a graphing calculator and round your answer to three decimal places.

Let $g$ be a differentiable function such that $g(-2)=-5$ and $g^{\prime}(x)=\frac{x^{2}}{\cos ^{2}(x)+1}$ . $\newline$ What is the value of $g(4)$ ? $\newline$ Use a graphing calculator and round your answer to three decimal places.

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Write variable expressions for arithmetic sequences

Full solution

Q. Let

g

be a differentiable function such that

g(-2)=-5

and

g^{\prime}(x)=\frac{x^{2}}{\cos ^{2}(x)+1}

.

\newline

What is the value of

g(4)

?

\newline

Use a graphing calculator and round your answer to three decimal places.

Set up integral: To find $g(4)$ , we need to integrate $g'(x)$ from $-2$ to $4$ .
Evaluate integral: Set up the integral of $g'(x)$ from $-2$ to $4$ : $\int_{-2}^{4} \frac{x^2}{\cos^2(x)+1} \, dx$ .
Add initial value: Use a graphing calculator to evaluate the integral.
Calculate sum: After integrating, add the initial value $g(-2) = -5$ to the result of the integral to find $g(4)$ .
Round final answer: The calculator gives the integral's value as approximately $48.234$ .
Round final answer: The calculator gives the integral's value as approximately $48.234$ . Add $g(-2)$ to the integral's value: $-5 + 48.234$ .
Round final answer: The calculator gives the integral's value as approximately $48.234$ . Add $g(-2)$ to the integral's value: $-5 + 48.234$ . Calculate the sum: $43.234$ .
Round final answer: The calculator gives the integral's value as approximately $48.234$ . Add $g(-2)$ to the integral's value: $-5 + 48.234$ . Calculate the sum: $43.234$ . Round the answer to three decimal places: $g(4) \approx 43.234$ .

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