If [.] represents greatest integer function, then _ 3 4 ^ [ x+ [ 4 x ] ] d x=

Step-by-step solution

Given integral I= _ 3 4 ^ x-1 [ 4 x ] d x We known that x lies between -1 to 1 and we can split the values of x between -1 to 0 and 0 to 1 . If 0 " x^ 1 then [ x]=0 for x [0, ] Then, aligned & I = _ 3 4 ^ [ x] d x+ _ 3 4 ^ [ 4 x ] d x & If 3 4 x then 4 x 3 4 4 4 & 3 4 x 4 aligned So, [ 4 x ]=3 aligned & I= _ 3 4 ^ (0) d x+3 _ 3 4 ^ d x & I=0+4[x]_ 3 4 ^ =3 ( - 3 4 )= 3 4 aligned