Hi!!!
seems easy...
Here it goes>>>

The given integral needs to be divided into separate limits.
Now,
since there is a factor of 2 within the braces, we need to divide it into limits such that within a pair of limits the function takes values between two integers.

Clearly,

_[/2]^[3/2] [2sinx]

 

=_[/2]^[5/6] [2sinx]dx + _[5/6]^[] [2sinx]dx +  _[]^[7/6] [2sinx]dx

 + _[7/6]^[3/2] [2sinx]dx

 

=_[/2]^[5/6] 1dx + _[5/6]^[] 0dx + _[]^[7/6] (-1)dx + _[7/6]^[3/2] (-2)dx

 

= 5/6 - /2 - 7/6 +  - 3 + 7/3

 

= -/2