`int(x^(e-1)+e^(x-1))/(x^(2)+e^(x))` dx is equal to

Options

A. `log(x^(e)+e^(x))+c`
B. `elog(x^(e)+e^(x))+c`
C. `(1)/(e)log(x^(e)+e^(x))+c`
D. None of the above

Answer

C

Explanation

Put `x^(e)+e^(x)=t`
`rArr e(x^(e-1)+e^(x-1))dx=dt`
`:. int (x^(e-x)+e^(x-1))/(x^(e)+e^(x)) dx=(1)/(e) int (dt)/(t)`
`=(1)/(e) log t+c`
`=(1)/(e) log (x^(e)+e^(x))+c`