InterviewSolution
Saved Bookmarks
InterviewSolution
| 1. |
Evaluate `int(dx)/((e^(x)-1)).` |
|
Answer» Putting `e^(x)=t and e^(x)dx=dt,i.e., dx=(1)/(t),`we get `I=int(dx)/((e^(x)-1))=int(dt)/(t(t-1)).` `Let(1)/(t(t-1))=(A)/(t)+(B)/(t(t-1)).` `Then,1-=A(t-1)+Bt.` Putting `t=0`in (i) ,we get `A=-1.` Putting `t=1`in (i) , we get `B=1.` `therefore (1)/(t(t-1))=(-1)/(t)+(1)/((t-1)).` `"Hence",I=int(dx)/((e^(x)-1))` `=int(dt)/(t(t-1))=int(-1)/(t)dt+int(1)/((t-1))dt` `=-log |t|+log|t-1|+C` `=log |(t-1)/(t)|+C` `=log |(e^(x)-1)/(e^(x))|+C` |
|