(dx)/(1-bar(e)^(2)u)=log(e^(x)+sqrt((e^(x))^(2)-e))+c

int (dx)/(sqrt(2e^(x)-1))=2 sec^(-1)(sqrt(2)e^((x)/(2)))+c

If int(e^(4x)-1)/(e^(2x))log((e^(2x)+1)/(e^(2x)-1))dx=(t^(2))/(2)logt-(t^(2))/(4)-(u^(2))/(2)logu+(u^(2))/(4)+C, then

If int(e^(4x)-1)/(e^(2x))log((e^(2x)+1)/(e^(2x-1)))dx=(t^(2))/(2)logt-(t^(2))/(4)-(u^(2))/(2)logu+(u^(2))/(4)+C then

intsqrt((e^x+1)/(e^x-1))dx (A) ln (e^(x)+sqrt(e^(2x)-1))-sec^(-1)(e^(x)) +C (B) ln(e^(x)+sqrt(e^(2x)-1))+sec^(-1)(e^(x))+C (C) ln (e^(x)-sqrt(e^(2x)-1))-sec^(-1)(e^(x)) +C (D) ln(e^(x)+sqrt(e^(2x)-1))-sin^(-1)(e^(-x))+C

int((1+(log)_(e)x)^(2))/(1+(log)_(e)x^(x+1)+((log)_(e)x^(sqrt(x)))^(2))dx=

int(e^(x)-e^(-x))/(e^(2)x+e^(-2)x)dx=A log|(e^(x)+e^(-x)+a)/(e^(x)+e^(-x)-a)|+c then (A,a) is (A)(-(1)/(2sqrt(2)),-sqrt(2))(B)(-(1)/(sqrt(2)),2sqrt(2))(C)((1)/(sqrt(2)),-2sqrt(2))(D)((1)/(2sqrt(2)),-sqrt(2))

int(1)/(1+e^(x))dx= (a) log(1+e^(x)) (b) log((1+e^(x))/(e^(x))) (c) log(1+e^(-x)) (d) -log(e^(-x)+1)

if int dx/(e^(x) (e^(x)+1)^(2))=(2e^(x)+1)/(e^(x)(e^(x)+1))+ k log|1+e^(-x)|+c then the value of k is -