Question

If $\int \frac{2x+3}{x^2-5x+6}dx=9ln(x-3)-7ln(x-2)+A$, then A=

• A. 5 ln (x-2) +constant
• B. -4 ln (x-3)+constant
• C. Constant
• D. None of these
  

Answer 1

The correct option is C Constant

$\int \frac{2x+3}{x^2-5x+6}dx=\int \frac{2x-5}{x^2-5x+6}dx+\int \frac{8}{x^2-5x+6}dx=log(x^2-5x+6)+8\int \frac{1}{(x-2)(x-3)}dx+c$.
$=\log \left[\right(x-2\left)\right(x-3\left)\right]+8\int [\frac{1}{x-3}-\frac{1}{x-2}]dx+c.=log(x-2)+log(x-3)+8log(x-3)-8log(x-2)+c.=9log(x-3)-7log(x-2)+c....\left(i\right)$
Now given that
$\int \frac{2x+3}{x^2-5x+6}dx=9log(x-3)-7(x-2)+A$
Equating it to (i), we get A= constant.