Answer 1

According to question,

let  I = `int_  (sin "x" - cos "x" )/sqrt(1 + sin 2"x") d"x", 0 < "x" < π / 2 `

I = `int_  (sin "x" - cos "x")/sqrt(sin^2 "x" + cos^2 "x" + 2 sin "x" .cos "x") d"x"`

= `int_  ( sin "x" - cos "x")/sqrt((sin"x" + cos "x")^2 d"x"`

= `int_  (sin "x" - cos "x")/(sin "x" + cos "x") d"x"`

let sin x + cos x = t

⇒ (cos x - sin x) dx = dt

I = `int_  (-1)/("t") d"t"`

= -ln t + C

= ln `(1/"t") + "C"`

⇒ I = ln `((1)/(sin "x" + cos "x")) + "C"`