Question

Evaluate: `int_((-pi)/6)^(pi/2)(sin|x|+cos|x|)dx`

Answer 1

Let I = `int_((-pi)/6)^(pi/2)(sin|x|+cos|x|)dx`

|x| = `{{:(x,ifx≥0),(-x,ifx<0):}`

I = `int_(-pi/6)^0[sin(-x)+cos(-x)]dx+int_0^pi/2[sin(x)+cos(x)]dx`

I = `int_((-pi)/6)^0[-sinx+cosx]dx+int_0^(pi/2)[sinx+cosx]dx`

I = `[cosx+sinx]_((-pi)/6)^0 +[-cosc+sinx]_0^(pi/2)`

= `[(cos0+sin0)-(cos((-pi)/6)+sin((-pi)/6))]+[(-cos  pi/2+sin  pi/2)-(-cos0+sin0)]`

= `[(1+0)-(sqrt3/2-1/2)]+[(0+1)-(-1+0)]`

= `[1-((sqrt3-1)/2)]+2`

= `(2-sqrt3+1+4)/2`

= `(7-sqrt3)/2`