The value of the integral `int_(a)^(a+pi//2) (|sin x|+|cosx|)dx` is

Options

A. `a pi`
B. `2 a pi`
C. `(a pi)/(2)`
D. indenpedent of a

Answer

D

Explanation

Since `f(x)=|sin x|+|cos x| `is a periodic function with period `(pi)/(2)`. Therefore, `underset(a)overset(a+pi//2) intf(x)dx` is indepdent of a . In fact, we have
`underset(a)overset(a+pi//2) intf(x)dx=underset(a)overset(a+pi//2) int f(x)dx =underset(a)overset(a+pi//2) int (|sin x|+|cos x |)dx`
`rArr underset(a)overset(a+pi//2) intf(x)dx=underset(a)overset(a+pi//2) int (sinx+cosx )dx`