Question

∫[dx/(e^x + 1)] = _________ + c 

(a) – log |[(e^x + 1)/(e^x)]| 

(b) – log |[(e^x)/(e^x + 1)]| 

(c) log |[(e^x + 1)/(2e^x)]| 

(d) log |[(e)^2x/(e^x + 1)]|

Answer 1

The correct option (a) – log |[(e^x + 1)/(e^x)]|  

Explanation:

I = ∫[dx/(e^x + 1)] 

= ∫[(e^–xdx)/(1 + e^–x)] 

Put 1 + e^–x = t 

∴ e^–x (− 1)dx = dt 

∴ I = ∫[(− dt)/t] 

= – logt + c 

∴ I = – log(1 + e^–x) + c 

= – log[1 + (1/e^x)] + c 

= – log[(e^x + 1)/e^x] + c