If the primitive of $$sin$$−$$3/2$$⁡xsin−$$1/2$$⁡$$(x+$$θ$$)is$$ −$$2cosec$$⁡θ$$f(x)+C$$ then

Question

If the primitive of $$sin$$−$$3/2$$⁡xsin−$$1/2$$⁡$$(x+$$θ$$)is$$ −$$2cosec$$⁡θ$$f(x)+C$$ then

Infinity Learn · Accepted Answer

The primitive of the given function is ∫$$dxsin3$$⁡xsin⁡(x+$$θ$$)=$$∫$$dxsin3$$⁡$$x($$$$sin$$⁡xcos⁡θ$$+$$$$cos$$⁡x∣$$sin$$⁡θ$$)=$$∫$$cosec2$$⁡xdxcos⁡θ$$+$$$$cot$$⁡xsin⁡θ$$=$$−$$1sin$$⁡θ∫dtcot⁡θ$$+t(t=$$$$cot$$⁡$$x)=$$−$$2sin$$⁡θ$$($$$$cot$$⁡θ$$+t)1/2+C=$$−$$2sin$$⁡θ$$($$$$cos$$⁡θ$$+$$$$sin$$⁡θ$$t)1/2+C=$$−$$2cosec$$⁡θ$$($$$$sin$$⁡xcos⁡θ$$+$$$$sin$$⁡θ$$cos$$⁡$$x)1/2$$ $$+Csin$$⁡x    $$=$$−$$2cosec$$⁡θ$$sin$$⁡$$($$θ$$+x)$$sin$$⁡$$x1/2+C$$.

If the primitive of $\sin^{- 3 / 2} x \sin^{- 1 / 2} (x + θ)$
is $- 2 cosec θ \sqrt{f (x)} + C$ then

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If the primitive of sin−3/2⁡xsin−1/2⁡(x+θ)is −2cosec⁡θf(x)+C then

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If the primitive of $\sin^{- 3 / 2} x \sin^{- 1 / 2} (x + θ)$
is $- 2 cosec θ \sqrt{f (x)} + C$ then