1.

If `I=int(e^x)/(e^(4x)+e^(2e)+1) dx. J=int(e^(-x))/(e^(-4x)+e^(-2x)+1) dx.` Then for an arbitrary constant c, the value of `J-I` equal toA. `(1)/(2)log((e^(4x)-e^(2x)+1)/(e^(4x)+e^(2x)+1))+C`B. `(1)/(2)log((e^(2x)+e^(x)+1)/(e^(2x)-e^(x)+1))+C`C. `(1)/(2)log((e^(2x)+e^(x)+1)/(e^(2x)+e^(x)+1))+C`D. `(1)/(2)log((e^(2x)+e^(2x)+1)/(e^(2x)+e^(2x)+1))+C`

Answer» Correct Answer - c
We have , `J-I=int(e^(3x))/(e^(4x)+e^(2x)+1)-(e^(x))/(e^(4)+e^(2x)+1)dx`
`rArrJ-I=int(e^(2x)-1)/(e^(4x)+e^(2x)+1)d(e^(x))`
`rArrJ-I=int(t^(2)-1)/(t^(4)+t^(2)+1)dt` where `t=e^(x)`
`rArrJ-I=int(1)/((t+(1)/(t))^(2)-1)d(t-(1)/(t))`
`rArrJ-I=(1)/(2)log((t+(1)/(t)-1)/(t+(1)/(t)+1))+C`
`rArrJ-I=(1)/(2)log((t^(2)-t+1)/(t^(2)+t+1))+C=(1)/(2)log((e^(2x)-e^(x)+1)/(e^(2x)+e^(x)+1))+C`


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