Question

$\int \left(\frac{4e^x-25}{2e^x-5}\right)dx=Ax+B\log |2e^x-5|+c$, then

• A. $A=5,B=3$
• B. $A=5,B=-3$
• C. $A=-5,B=3$
• D. $A=-5,B=-3$

Answer 1

Answer: (B)

$A=5,B=-3$

Given : $\int \left(\frac{4e^x-25}{2e^x-5}\right)dx=Ax+B\log |2e^x-5|+c$

$⟹\int \left(\frac{10e^x-6e^x-25}{2e^x-5}\right)dx=Ax+B\log |2e^x-5|+c$

$⟹\int (5-\frac{6e^x}{2e^x-5})dx=Ax+B\log |2e^x-5|+c$

$⟹\int 5.dx-3\int \left(\frac{2e^x}{2e^x-5}\right)dx=Ax+B\log |2e^x-5|+c$

$⟹5x-3\log |2e^x-5|+c=Ax+B\log |2e^x-5|+c$

Equating the like terms we get

$A=5$ and $B=-3$

Hence, option B is correct.