Question

If $\int \frac{3e^x-5e^{-x}}{4e^x+5e^{-x}}dx=ax+b\ln (4e^x+5e^{-x})+C$, then

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

Answer 1

The correct option is A $a=-\frac{1}{8},b=\frac{7}{8}$

$\int \frac{3e^x-5e^{-x}}{4e^x+5e^{-x}}=ax+b\ln (4e^x+5e^{-x})+C$

Differentiating both sides, we get
$\frac{3e^x-5e^{-x}}{4e^x+5e^{-x}}=a+b\frac{(4e^x-5e^{-x})}{4e^x+5e^{-x}}$

$3e^x-5e^{-x}=a(4e^x+5e^{-x})+b(4e^x-5e^{-x})$

Comparing the coefficients of like terms on both sides, we get
$3=4(a+b),-5=5a-5b$

$\Rightarrow a=-\frac{1}{8},b=\frac{7}{8}$