`int_((pi)/4)^((pi)/2)(2cosecx)^17` dx

Options

A. `int_(0)^(log(1+sqrt(2)))2(e^(u)+e^(-u))^(16)du`
B. `int_(0)^(log(1+sqrt(2))) 2(e^(u)+e^(-u))^(17)du`
C. `int_(0)^(log(1+sqrt(2))) 2(e^(u)-e^(-u))^(17)du`
D. `int_(0)^(log(1+sqrt(2)))2(e^(u)-e^(-u))^(16)du`

Answer

A

Explanation

`int_((pi)/4)^((pi)/2) (2cosecx)^(17)dx`
Let `e^(u)+e^(-u)=2cosecx`,
For `x=(pi)/4,u=In(1+sqrt(2))`
For `x=(pi)/2,u=0`
Also `cosecx+cotx=e^(u)` and `cosecx-cotx=e^(-u)`
`impliescotx=(e^(u)-e^(-u))/2`
Also `(e^(u)-e^(-u))du=-2cosec x cot x dx`
`implies I=-int_(In(1+sqrt(2)))^(0)(e^(u)+e^(-u))^(17)((e^(u)-e^(-u)))/(2cosecxcotx)du`
`=-2int_(In(1+sqrt(2)))^(0)(e^(u)+e^(-u))^(16)du`
`int_(0)^(In(1+sqrt(2)))2(e^(u)+e^(-u))^(16)du`