Evaluate `intsqrt(1+"cosec"x)dx,(pi//2lt x lt pi)`

Answer

`2"sin"^(-1)("sin"(x)/(2)-"cos"(x)/(2))+c`

Explanation

`I=intsqrt(1+"cosec"x)dx`
`=int(sqrt(1+sinx))/(sqrt(sinx))dx`
`=int("sin"(x)/(2)+"cos"(x)/(2))/(sqrt(2"sin"(x)/(2)" cos"(x)/(2)))dx " "( :' 0lt x lt pi//2)`
`=int("sin"(x)/(2)+"cos"(x)/(2))/(sqrt(1-("sin"(x)/(2)-"cos"(x)/(2))^(2)))dx`
` "Put sin"(x)/(2)="cos"(x)/(2)=t " or " ("cos"(x)/(2)+"sin"(x)/(2))dx=2dt`
` :. I=int(2dt)/(sqrt(1-t^(2)))=2"sin"^(-1)t+c`
`=2"sin"^(-1)("sin"(x)/(2)-"cos"(x)/(2))+c`