Chapter 5 - Section 5.4 - The Fundamental Theorem of Calculus - Exercises - Page 322: 79

$-3x+5$

By the Fundamental Theorem of Calculus, the function $f(x)$ is the derivative of $f(x)=2-\int_2^{x+1} \dfrac{9}{1+t}dt$ is equal to $f'(x)=\dfrac{-9}{1+x+1}$ Thus,$f(1)=2; f'(1)=-3$ The linearization of $f(x)$ at $x=1$ is$L(x)=f(1)+f'(1)(x-1)$ Now, $L(x) =2-3(x-1)=-3x+5$