If `f(x)=|(sin x+sin x 2x+sin3x,sin2x,sin3x),(3+4sinx,3,4sinx),(1+sinx,sinx,1)|,` then the value of `int_0^(pi/2) f(x) dx` is

Options

A. 3
B. `(2)/(3)`
C. `(1)/(3)`
D. 0

Answer

C

Explanation

`f(x)=|{:(sinx+sin2xsin3x,sin2x,sin3x),(3+4sinx,3,4sinx),(1+sinx,sinx,1):}|`
[Applying `C_(1)toC_(1)-(C_(2)+C_(3))]`
`=|{:(sinx,sin2x,sin3x),(0,3,4sinx),(0,sinx,1):}|`
`=sinx(3-4sin^(2)x)=sin3x`
Then , `int_(0)^(pi//2)f(x)dx=int_(0)^(pi//2)sin3xdx=-[(cos3x)/(3)]_(0)^(pi//2)`
`=(-(1)/(3))[cos(3xx(pi)/(2))-cos0]`
`=(-(1)/(3))(0-1)=(1)/(3)`