Question

$\int \frac{e^x}{(e^x+2)(e^x-1)}dx=$

• A. $\frac{1}{3}\log |\frac{e^x-1}{e^x+2}|+c$
• B. $\frac{1}{3}\log |\frac{e^x+1}{e^x-1}|+c$
• C. $\frac{1}{3}\log |\frac{e^x+1}{e^x+2}|+c$
• D. $-\frac{1}{3}\log |\frac{e^x+1}{e^x+2}|+c$

Answer 1

The correct option is A $\frac{1}{3}\log |\frac{e^x-1}{e^x+2}|+c$

$\int \frac{e^x}{(e^x+2)(e^x-1)}dx$
$let{e}^x=t$
$e^{x}dx=dt$
$\int \frac{dt}{(t+2)(t-1)}=\frac{1}{3}\int \frac{(t+2)-(t-1)}{(t+2)(t-1)}dt$
$=\frac{1}{3}[\int \frac{1}{(t-1)}dt-\int \frac{1}{(t+2)}dt]$
$=\frac{1}{3}\left[log\right(t-1)-log(t+2\left)\right]+c$
$=\frac{1}{3}log\left(\left|\frac{t-1}{t+2}\right|\right)+c$
$=\frac{1}{3}log\left|\frac{e^x-1}{e^x+2}\right|+c$