Statement -1 : If `I_(1)=int(e^(x))/(e^(4x)+e^(2x)+1)dx` and
`I_(2)=int(e^(-x))/(e^(-4x)+e^(-2x)+1)dx`, then
`I_(2)-I_(1)=(1)/(2)log((e^(2x)-e^(x)+1)/(e^(2x)+e^(x)+1))+C`
where C is an arbitrary constant.
Statement -2 : A primitive of f(x) `=(x^(2)-1)/(x^(4)+x^(2)+1)` is
`(1)/(2)log((x^(2)-x+1)/(x^(2)+x+1))`.

Options

A. Statement - 1 True , Statement -2 is True , Statement -2 is a correct explanation for Statement -1.
B. Statement - 1 is True , Statement -2 is True , Statement -2 is a correct explanation for Statement -1.
C. Statement - 1 True ,Statement - 2 is False.
D. Statement - 1 is False , Statement - 2 is True.

Explanation

A primitive of f (x) `=(x^(2)-1)/(x^(4)+x^(2)+1)` is given by
`I=int(x^(2)=1)/(x^(4)+x^(2)+1)dx=int(1-(1)/(x^(2)))/(x^(2)+(1)/(x^(2))+1)dx`
`rArrI=int(1)/((x+(1)/(x))^(2)-1^(2))d(x+(1)/(x))=(1)/(2)log|(x+(1)/(x)-1)/(x+(1)/(x)+1)|+C`
`rArrI=int(1)/(2)log((x^(2)-x+1)/(x^(2)+x+1))+C`
So , statement - 2 is true.
Now , `I_(2)-I_(1)=int(e^(-x))/(e^(-4x)+e^(-2x)+1)dx-int(e^(x))/(e^(4x)+e^(2x)+1)dx`
`rArrI_(2)-I_(1)=int(e^(3x))/(e^(4x)+e^(2x)+1)dx-int(e^(x))/(e^(4x)+e^(2x)+1)dx`
`rArrI_(2)-I_(1)=int(e^(3x))/(e^(4x)+e^(2x)+1)d(e^(x))`
`rArrI_(2)-I_(1)=(1)/(2)log((e^(2x)-e^(x)+1)/(e^(2x)+e^(x)+1))+C` [ Using statment -2 ]
So , statement - 1 is true . Also . statement - 2 is a correct explanation for statement -1.