Question

$\int \frac{adx}{b+c{e}^x}=$

• A. $\frac{a}{b}log\left(\frac{e^x}{b+c{e}^x}\right)+c$
• B. $\frac{a}{b}log\left(\frac{b+c{e}^x}{e^x}\right)+c$
• C. $\frac{b}{a}log\left(\frac{e^x}{b+c{e}^x}\right)+c$
• D. $\frac{b}{a}log\left(\frac{b+c{e}^x}{e^x}\right)+c$

Answer 1

The correct option is A $\frac{a}{b}log\left(\frac{e^x}{b+c{e}^x}\right)+c$

$\int \frac{adx}{b+c{e}^x}=\int \frac{a{e}^x}{b{e}^x+c{e}^{2x}}dx$
Now put $e^x=t$, then it reduces to
$a\int \frac{dt}{t(ct+b)}=a\int -\frac{1}{b}\{\frac{c}{ct+b}-\frac{1}{t}\}dt$ {By partial fraction}
$=\frac{a}{b}log\left(\frac{e^x}{b+c{e}^x}\right)+c$