Question

Integrate the following functions w.r.t. x.

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

Answer 1

$LetI=\int \frac{e^x}{(1+e^x)(2+e^x)}dxPut{e}^x=t\Rightarrow e^{x}dx=dt\therefore I=\int \frac{e^x}{(1+t)(2+t)}\frac{dt}{e^x}=\int \frac{1}{(1+t)(2+t)}dtLet\frac{1}{(1+t)(2+t)}=\frac{A}{(1+t)}+\frac{B}{(2+t)}\Rightarrow 1=A(2+t)+B(1+t)=(2A+B)+t(A+B)$

On equating the coefficients of t and constant term on both sides , we get A + B = 0 and 2A + B = 1

On solving both equations, we get A = 1 and B = -1

$\therefore I=\int \frac{1}{(1+t)}dt-\int \frac{1}{(2+t)}dt=log|1+t|-log|2+t|+C=log\left|\frac{1+t}{2+t}\right|+C=log\left|\frac{1+e^x}{2+e^x}\right|+C$