Evaluate: `int_(pi//6)^(pi//4)(1+cotx)/(e^(x)sinx) dx`

Answer

`2e^(-pi//6)-sqrt(2)e^(-pi//4)`

Explanation

`I=int_(pi//6)^(pi//4) e^(-x)(cosecx+cot x cosec x)dx`
Put `x=timpliesdx=-dt`
`:. I=-int_(-pi//6)^(-pi//4) e^(t)(-cosect+cot.cosec)dt`
`=int_(-pi//6)^(-pi//4)e^(t)(cosect-cot t. cosec t)dt`
`=e^(t)cosec t|_(-pi//6)^(-pi//4)`
`=-sqrt(2)e^(-pi//4)+2d^(-pi//6)`
`=2e^(-pi//6)-sqrt(2)e^(-pi//4)`