Evaluate: `int_(-(3pi)/2)^(-pi/2)[(x+pi)^3+cos^2(x+3pi)]dx`

Options

A. `(pi^(4))/(32)`
B. `(pi^(4))/(32)+(pi)/(2)`
C. `(pi)/(2)`
D. `(pi)/(4)-1`

Answer

C

Explanation

`I=overset(-pi//2)underset(-3pi//2)int{(x+pi)^(3)+cos^(2)(x+3pi)}dx`
`rArrI=overset(pi//2)underset(-pi//2)int{t^(3)+cos^(2)(2pi+t)}dt`
`rArr I=overset(pi//2)underset(-pi//2)int t^(3)dt+overset(pi//2)underset(-pi//2)int cos^(2)t dt=0+2overset(pi//2)underset(0)int cos^(2)t dt =2xx(pi)/(4)=(pi)/(2)`