Question

if $\int \frac{2x+3}{x^2-5x+6}dx=9\log |x-3|\cdot 7\log |x-2|+A$ then $A=$

• A. $5\log (x+2)+constant$
• B. $-4\log (x-3)+constant$
• C. constant
• D. None of these

Answer 1

Answer: (C)

constant

$\int \frac{2x+3}{x^2-5x+6}dx=9\log |x-3|-7\log |x-2|+A$

$A=?$

$\int \frac{2x+3}{x^2-5x+6}dx=\int \frac{2x+3}{(x-3)(x-2)}dx=\int 2x+3[\frac{1}{(x-3)}-\frac{1}{x-2}]dx$

$=\frac{2x+3}{x-3}dx-\int \frac{2x+3}{x-2}dx$

$=\int (2+\frac{9}{x-3})dx-(\int (2+\frac{7}{x-3})dx)=2x+9\log (x-3)-2x-7\log (x-3)+c$

$=9\log (x-3)-7\log (x-3)+c$

So: A = constant