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${:[g(x)=int_(2)^(x)(13)/(1+t^(2))dt],[g^(')(5)=]:}$

$\begin{array}{l}g(x)=\int_{2}^{x} \frac{13}{1+t^{2}} d t \\ g^{\prime}(5)=\end{array}$

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Find indefinite integrals using the substitution

Full solution

Q.

\begin{array}{l}g(x)=\int_{2}^{x} \frac{13}{1+t^{2}} d t \\ g^{\prime}(5)=\end{array}

Apply Fundamental Theorem of Calculus: We use the Fundamental Theorem of Calculus which says that if $g(x) = \int_{a}^{x}f(t)\,dt$ , then $g'(x) = f(x)$ .
Calculate $g'(x)$ : So, $g'(x) = \frac{13}{1+x^{2}}$ .
Find $g'(5)$ : Now we need to find $g'(5)$ , which means we plug in $x=5$ into $g'(x)$ .
Substitute $x=5$ : $g'(5) = \frac{13}{1+5^{2}}$ .
Simplify Result: Calculate $g'(5) = \frac{13}{1+25}$ .
Simplify Result: Calculate $g'(5) = \frac{13}{1+25}$ . $g'(5) = \frac{13}{26}$ .
Simplify Result: Calculate $g'(5) = \frac{13}{1+25}$ . $g'(5) = \frac{13}{26}$ . Simplify $g'(5) = \frac{1}{2}$ .

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