Question

$\int \frac{3 e ^{ x }+5 e ^{- x }}{4 e ^{ x }-5 e ^{- x }} dx = Ax + B \ln \left|4 e ^{2 x }-5\right|+ C$ then

• A. $A =-1, B =-7 / 8 ; C =$ const. of integration
• B. $A =1, B =7 / 8 ; C =$ const. of integration
• C. $A =-1 / 8, B =7 / 8 ; C =$ const. of integration
• D. $A =-1, B =7 / 8 ; C =$ const. of integration

Answer 1

Answer: (D)

$ I=\int \frac{3 e^x+5 e^{-x}}{4 e^x-5 e^{-x}} d x=\int \frac{3 e^{2 x}+5}{4 e^{2 x}-5} d x$
Let $3 e ^{2 x }+5= A \left(4 e ^{2 x }-5\right)+ B \left(8 e ^{2 x }\right)$
$\therefore 4 A+8 B=3 $
$\text { and } 5 A=5 \Rightarrow A=-1$
and $5 A =5 \Rightarrow A =-1$
$\therefore B =\frac{7}{8}$
$\therefore I = Ax + B \ln \left(4 e ^{2 x }-5\right)+ C \text { when } A =-1 \text { and } B =\frac{7}{8}$