If In=∫cosecn⁡xdx then 9I10−8I8 is equal to

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

If In=∫cosecn⁡xdx then 9I10−8I8 is equal to

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

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$- \cos e c^{8} x \cot x + C$

$\cos e c^{8} x \cot x + C$

$\cos e c^{9} x \cot x + C$

$- \cos e c^{9} x \cot x + C$

answer is A.

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Detailed Solution

$I_{10} = \int \cos e c^{10} x d x$

$\begin{array}{l} = \int {cosec}^{8} x {cosec}^{2} x d x \\ = \cos e c^{8} x (- \cot x) - \int 8 {cosec}^{7} x (- cosec x \cot x) (- \cot x) d x \\ = - {cosec}^{8} x \cot x - 8 \int {cosec}^{8} x ({cosec}^{2} x - 1) d x \\ I_{10} = - {cosec}^{8} x \cot x - 8 \int {cosec}^{10} x d x + 8 \int {cosec}^{8} x d x \\ = - {cosec}^{8} x \cot x - 8 I_{10} + 8 I_{8} \\ 9 I_{10} - 8 I_{8} = - {cosec}^{8} x \cot x + C \end{array}$