integration of 1/√sin^3 x. cos(x-a)
Answers
Given : 1/√sin³x. cos(x-a)
To Find : Integrate
Solution:
I =∫ 1/√sin³x. cos(x-a) dx
1/√sin³x. cos(x-a)
= 1/√sin³x. (cosxcosa + sinxsina)
= 1/√sin³x. sinx(cotxcosa + sina)
= 1/√sin⁴x(cotxcosa + sina)
= 1/sin²x.√(cotxcosa + sina)
= cosec²x/√(cotxcosa + sina)
I = ∫(cosec²x/√(cotxcosa + sina)) dx
√(cotxcosa + sina) = t
=> (1/2√(cotxcosa + sina))(-cosec²xcosa) dx = dt
=> (cosec²x/√(cotxcosa + sina)) dx = -2dt/cosa
I = ∫-2dt/cosa
=> I = - (2/cosa)∫dt
=> I = - (2/cosa)t + C
=> I = - 2t/Cosa + C
substitute value of t
=> I = - 2√(cotxcosa + sina) / cosa + C
√(cotxcosa + sina) = √(cos(x -a)/sinx)
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