If ₀^ d x 1+2 ^2 x =k , then greatest integer less than or equal to k is

Step-by-step solution

aligned & ₀^ 1 1+2 ^2 x d x=k & 2 ₀^ / 2 1 1+2 ^2 x d x & [ 1 1+2 ^2 x is an even function ] & 2 ₀^ / 2 ^2 x 1+ ^2 x+2 ^2 x d x & 2 ₀^ / 2 ^2 x 1+3 ^2 x d x & Let x=t & x d x=d t & When x 0, t 0 aligned and when x / 2, t aligned & 2 ₀^ 1 1+3 t^2 d t & 2 3 [ ⁻¹ t 3 ]₀^ & 2 3 ( 2 -0 ) & 3 = [ 3 ]= [ 3.14 1.73 ]=1 aligned