Chapter 5: Integrals - Section 5.4 - The Fundamental Theorem of Calculus - Exercises 5.4 - Page 288: 69

$-3x+5$

The function $f(x)$ is the derivative of $f(x)=2-\int_2^{x+1} \dfrac{9}{1+t}dt$ is equal to $f'(x)=\dfrac{-9}{1+x+1}$ by the fundamental Theorem of calculus. Here, we have $f(1)=2; f'(1)=-3$ The linearization of $f(x)$ at $x=1$ is given as: $L(x)=f(1)+f'(1)(x-1)$ $\implies L(x) =2-3(x-1)$ Thus, $L(x) =-3x+5$