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