Question

If $\int \frac{3e^x-5e^{-x}}{4e^x+5e^{-x}}dx=Ax+B\ln |4e^x+5e^{-x}|+C,$ then which of the following is/are true
(where $A,B$ are fixed constants and $C$ is constant of integration)

• A. $A=-\frac{1}{8}$
• B. $A=\frac{1}{8}$
• C. $B=\frac{7}{8}$
• D. $B=-\frac{7}{8}$

Answer 1

Answer: (A), (C)

$B=\frac{7}{8}$

$\int \frac{3e^x-5e^{-x}}{4e^x+5e^{-x}}dx=Ax+B\ln (4e^x+5e^{-x})+C$
Differentiating w.r.t. $x$ on both sides, we get
$\frac{3e^x-5e^{-x}}{4e^x+5e^{-x}}=A+B\frac{(4e^x-5e^{-x})}{(4e^x+5e^{-x})}$
$\Rightarrow 3e^x-5e^{-x}=A(4e^x+5e^{-x})+B(4e^x-5e^{-x})$ Comparing the coefficients of terms containing $e^x$ and $e^{-x}$ on both sides, we get $4(A+B)=3\Rightarrow A+B=\frac{3}{4}$
$5A-5B=-5\Rightarrow A-B=-1$
Solving these equations, we get
$A=-\frac{1}{8},B=\frac{7}{8}$ ​​​​​​