Question

Bài 5 [M2]. Biết rằng, với \( x \in (2; +\infty) \), \( F'(x) = a \ln x + b \ln (x-2) \) là một nguyên hàm của \( f(x) = \frac{2x+3}{x^2-2x} \) (a, b là các hằng số). Khi đó, a + b bằng

A. 0. \hspace{1cm} B. 1. \hspace{1cm} C. 2. \hspace{1cm} D. 3.

Answer 1

\(f\left(x\right)=\dfrac{2x+3}{x^2-2x}=\dfrac{2x+3}{x\left(x-2\right)}=\dfrac{A}{X}+\dfrac{B}{X-2}\)

=>\(A\left(X-2\right)+BX=2x+3\)

=>\(X\left(A+B\right)-2A=2x+3\)

=>\(\left\{{}\begin{matrix}A+B=2\\-2A=3\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}A=-\dfrac{3}{2}\\B=2-A=2-\left(-\dfrac{3}{2}\right)=\dfrac{7}{2}\end{matrix}\right.\)

=>\(f\left(x\right)=\dfrac{-\dfrac{3}{2}}{x}+\dfrac{\dfrac{7}{2}}{x-2}\)

=>\(F\left(x\right)=\int f\left(x\right)dx=-\dfrac{3}{2}\cdot lnx+\dfrac{7}{2}\cdot ln\left(x-2\right)\)

=>a=-3/2;b=7/2

a+b=-3/2+7/2=4/2=2