Evaluate the following integral:`int_(-3pi//2)^(-pi//2){sin^2(3pi+x)+(pi+x)^3}dx`

Explanation

Given: `int_(-(3pi)/2)^(-pi/2){sin^2(3pi+x)+(pi+x)^3}dx`

`I=int_(-(3pi)/2)^(-pi/2){sin^2(3pi+x)+(pi+x)^3}dx`

`I=int_(-(3pi)/2)^(-pi/2) (pi+x)^3dx+int_(-(3pi)/2)^(-pi/2)(1-cos2(3pi+x))/2dx`

`I=([(pi+x)^4]_(-(3pi)/2)^(-pi/2))/4+1/2[x]_(-(3pi)/2)^(-pi/2)-1/2([sin2(3pi+x)]_(-(3pi)/2)^(-pi/2))/(2)`

`I=pi/2-1/4[0-0]`

`I=pi/2`