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$If I_{n} = \int {cosec}^{n} x d x then 9 I_{10} - 8 I_{8} is equal to$

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a

−cosec8xcotx+C

b

cosec8xcotx+C

c

cosec9xcotx+C

d

−cosec9xcotx+C

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detailed solution

Correct option is A

I10=∫cosec10xdx=∫cosec8⁡xcosec2⁡xdx=cos⁡ec8x(−cot⁡x)−∫8cosec7⁡x(−cosec⁡xcot⁡x)(−cot⁡x)dx=−cosec8⁡xcot⁡x−8∫cosec8⁡xcosec2⁡x−1dxI10=−cosec8⁡xcot⁡x−8∫cosec10⁡xdx+8∫cosec8⁡xdx=−cosec8⁡xcot⁡x−8I10+8I89I10−8I8=−cosec8⁡xcot⁡x+C

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