Evaluate: `int_0^((5pi)/(12))[tanx]dx ,w h e r e[dot]` deontoes the greatest integer function.

Explanation

Let `I=int_(0)^(5pi//12)[tanx]dx`.
Here `y=tanx` is a monotonically increasing function.
Also, when `x=0, tanx=0` and when `x=(5pi)/12, tan x=2+sqrt(3)`.
Hence `[tanx]` is discontinuous when `tanx=1, tanx=2, tanx=3`.
Thus, `x=tan^(-1)1,x=tan^(-1)2,x=tan^(-1)3`
`:. I=int_(0)^(tan^(-1)1)[tanx]dx+int_(tan^(-1))^(tan^(-1)2)[tanx]dx`
`+int_(tan^(-1)2)^(tan^(-1)3)[tanx]dx+int_(tan^(-1)3)^(5pi//12)[tanx]dx`
`=int_(0)^(tan^(-1)1) 0 dx +int_(tan^(-1)1)^(tan^(-1)2)1dx+int_(tan^(-1)2)^(tan^(-1)3)2 dx+int_(tan^(-1)3)^(5pi//12)3dx`
`=0+(tan^(-1)2-tan^(-1))+2(tan^(-1)3-tan^(-1)2)`
`+3((5pi)/12-tan^(-1)3)`
`=(5pi)/4-(pi)/4-tan^(-1)3-tan^(-1)2`
`=pi-[pi+tan^(-1)((3+2)/(1-(3)(2)))]`
`=pi-[pi+tan^(-1)(-1)]=pi/4`