" 14."(e^(x)(x^(2)+1))/((x+1)^(2))

(e^(2x)+2e^(x)+1)/(e^(x))

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)) .

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)) .

int e^(x)((x-2)/((x-1)^(2)))dx is equal to (i) (e^(x))/(x-1)+C (ii) (e^(x))/((x-1)^(2))+C( iii) (2e^(x))/((x-1)^(2))+C( iv )(e^(x)-1)/(x-1)+C

e^(x)((x - 1)/(2x^(2)))

f(x)=(e^(2x)-1)/(e^(2x)+1) is

int(2e^(5x)+e^(4x)-4e^(3x)+4e^(2x)+2e^(x))/((e^(2x)+4)(e^(2x)-1)^(2))dx= a) "tan"^(-1)(e^(x))/(2)-(1)/(e^(2x)-1)+C b) "tan"^(-1)e^(x)-(1)/(2(e^(2x)-1))+C c) "tan"^(-1)(e^(x))/(2)-(1)/(2(e^(2x)-1))+C d) 1-"tan"^(-1)((e^(x))/(2))+(1)/(2(e^(2x)-1))+C

If y = (e^(x)-e^(-x))/(e^(x)+e^(-x)) then prove that y = (e^(2x)-1)/(e^(2x)+1) .

int_(1)^(2)((x^(2)-1)/(x^(2)))e^(x+(1)/(x))dx=e^((5)/(2))-e^(2)