Evaluate`intsqrt(secx-1)dx, 0ltxltpi/2`

Explanation

` I=int sqrt(secx-1)dx=int sqrt((1-cosx)/(cosx))dx`
`=int sqrt(((1-cosx))/(cosx)xx((1+cosx))/((1+cosx)))dx`
`=int sqrt((1-cos^(2)x)/(cosx+cos^(2)x))dx`
`=int(sinx)/(sqrt(cos^(2)x+cosx))dx`
Let `cosx=t.` Then `d(cosx)=dt` or `-sinx dx=dt.` Therefore,
`I=int(-dt)/(sqrt(t^(2)+t))`
`=-int(dt)/(sqrt((t+(1)/(2))^(2)-((1)/(2))^(2)))`
`=-log|(t+(1)/(2))+sqrt((t+(1)/(2))^(2)-((1)/(2))^(2))|+C`
` =-log|(t+(1)/(2))+sqrt(t^(2)+t)|+C`
`=-log|(cosx +(1)/(2))+sqrt(cos^(2)x+cosx)|+C`