Question

Solve $\int \frac{a}{b+c{e}^x}dx$

Answer 1

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

putting $b+c{e}^x=t$

$dx=\frac{dt}{c{e}^x}$

$=\int \frac{a}{t}\times \frac{dt}{t-b}$

$=a\int \frac{dt}{t(t-b)}$

$=a\int (\frac{-\frac{1}{b}}{t}-\frac{\frac{1}{b}}{t-b})$

$=\frac{a}{b}\int (\frac{1}{t-b}-\frac{1}{t})dt$

$=\frac{a}{b}[\log |t-b|-\log |t|+C]$

$=\frac{a}{b}\log |\frac{t-b}{t}|+C$

$=\frac{a}{b}\log |\frac{c{e}^x}{b+c{e}^x}|+C$