Question

Find `int dx/sqrt(sin^3x cos(x - α))`.

Answer 1

Let I = `int dx/sqrt(sin^3x cos(x - α))`

= `int dx/sqrt((sin^4x)/sinx [cosx cosα + sinx sinα]`

= `int dx/(sin^2 xsqrt(cotx cosα + sinα)`

= `int ("cosec"^2x  dx)/sqrt(cotx cosα + sinα)`

Let cot x cos α + sin α = t

Then, dt = – cosec² x cos α dx

∴ I = `int (-dt)/(cosαsqrt(t))`

= `(-2sqrt(t))/cosα + C`

= `(-2sqrt(cotx cosα + sinα))/cosα + C`

= `- 2 sec αsqrt(cotx cosα + sinα) + C`.