`int (dx)/(e^(x)+1-2e^(-x))` का मान है

Options

A. `log(e^(x)-1)-log(e^(x)+2)+C`
B. `(1)/(2)log(e^(x)-1)-(1)/(3)log(e^(x)+2)+C`
C. `(1)/(3)log(e^(x)-1)-(1)/(3)log(e^(x)+2)+C`
D. उपरोक्त में से कोई नहीं

Answer

C

Explanation

`int (dx)/(e^(x)+1-2e^(-x))= int (e^(x)dx)/((e^(x))^(2)+e^(x)-2)`
माना `e^(x)= t rArr e^(x)dx=dt`
`:. I= int (dt)/(t^(2)+t+1//4-2-1//4)`
`= int (dt)/((t+(1)/(2))^(2)-((3)/(2))^(2))`
`=(1)/(3)"log"[(t+1//2-3//2)/t+1//2+3//2]+C`
`=(1)/(3)"log" [(t-1)/(t+2)]+C`
`=(1)/(3)log(e^(x)-1)-(1)/(3)log(e^(x)+2)+C`