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$\int \frac{e^{\cot x}}{\sin^{2} x} (2 \log (\cos e c x) + \sin 2 x) d x$

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2ecotxlogcosecx+C

ecotxlogcosecx+C

−2ecotxlogcosecx+C

−2ecotxlogcosx+C

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detailed solution

Correct option is C

cot⁡x=t cosec2⁡xdx=dt−∫et2log⁡t2+1+2t1+t2dt=−etlog⁡t2+1+C=−ecot⁡xlog⁡1+cot2⁡x+C=−2ecot⁡xlog⁡(cos⁡ecx)+C

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∫ecotxsin2x2logcosecx+sin2xdx

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$\int \frac{e^{\cot x}}{\sin^{2} x} (2 \log (\cos e c x) + \sin 2 x) d x$