Question

Integrate the function:

`e^x/((1+e^x)(2+e^x))`

Answer 1

Let `I = e^x/((1 + e^x)(2 + e^x))`

Put e^x = t

e^x dx = dt

∴ I = `int dt/((1 + t)(2 + t))`

Let `1/((1 + t)(2 + t)) = A/(1 + t) + B/(2 + t)`

`=> 1 = A(2 + t) + B(1 + t)`     ....(1)

Putting t = -1 in equation (1),

∴ 1 = A(2 - 1)

⇒ A = 1

Putting t = -2 in equation (1),

∴ 1 = B(1 - 2)

⇒ B = - 1

`therefore 1/((1 + t)(2 + t))  dt`

`= int (1/(1 + t) - 1/(2 + t))  dt`

`therefore I = int 1/(1 + t) dt - int 1/(2 + t)  dt`

`= log |1 + t| - log |2 + t| + C`

`= log |1 + e^x| - log |2 + e^x| + C`

`= log |(1 + t)/(2 + t)| + C`

`= log |(1 + e^x)/(2 + e^x)| + C`