` int _(-3pi//2)^(-pi//2) [ ( x + pi)^(3) + cos^(2) x ] ` dx is equalt to

Options

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

Answer

B

Explanation

Let ` l = int_(-3pi//2)^(-pi//2)[(x+pi)^(3)+cos^(2)x]dx` … (i)
`rArrl = int_(-3pi//2)^(-pi//2)`
`[(-(pi)/(2)-(3pi)/(2)-x+pi)^(3)+cos^(2)(-(pi)/(2)-(3pi)/(2)-x)]dx`
`rArr l = int_(-3pi//2)^(-pi//2)[-(x+pi)^(3)+cos^(2)x]dx` … (ii)
On adding Eqs . (i) and (ii) , we get
`2l=int_(-3pi//2)^(-pi//2)2cos^(2)x dx=int_(-3pi//2)^(-pi//2)(1+cos2x)dx`
`=[x+(sin2x)/(2)]_(-3pi//2)^(-pi//2)`
`=[-(pi)/(2)+(sin(-pi))/(2)-(-3pi)/(2)+(sin(-3pi))/(2)]`
`rArrl=(pi)/(2)`