If f(t)= ₀^ 2 x ~d x 1- ^2 t ^2 x , 0 < t < , then the value of ₀^ 2 ^2 dt f( t ) equals_________

Step-by-step solution

f(t)= ₀^ 2 x 1- ^2 t ^2 x d x .....(1) =2 ₀^ ( -x) d x 1- ^2 ^2 x .....(2) aligned & 2 f(t)=2 ₀^ 1- ^2 ^2 x d x & f(t)= ₀^ 1- ^2 t ^2 x d x aligned divide & by ^2 x aligned & f(t)= ₀^ ^2 x d x ^2 x- ^2 t^2 x & f(t)=2 ₀^ / 2 ^2 x d x ^2 x- ^2 t^2 ^2 x & x=z & ^2 x d x=d z & f(t)=2 ₀^ d z 1+ ^2 t z^2 & = ^2 t aligned Then ₀^ / 2 ^2 f(t) d t aligned & = ₀^ / 2 t d t & =1 aligned