Question

$\int \frac{a}{b+c{e}^x}dx$ is equal to

• A. $\frac{a}{b}\log |b+c\cdot e^x|+K$
• B. $-\frac{a}{b}\log |c+b{e}^{-x}|+K$
• C. $\log |b+c\cdot e^x|+K$
• D. $\log |c+b\cdot e^{-x}|+K$

Answer 1

The correct option is B $-\frac{a}{b}\log |c+b{e}^{-x}|+K$

multiplying numerator and denominator with $e^{-x}$
$\int \frac{a{e}^{-x}}{b{e}^{-x}+c}dx$
Let $t=b{e}^{-x}+c,dt=-b{e}^{-x}dx$
Then
$I=\frac{-a}{b}\int \frac{dt}{t}$