## amurciano2 one year ago Integrate cot^5(x) csc^8(x)

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1. anonymous

$\cot ^5x=\cot x \cot ^4x=\cot x \left( \cot ^2x \right)^2=\cot x \left( \csc ^2x-1 \right)^2$ $=\cot x \left( \csc ^4x-2\csc ^2x+1 \right)$ $\csc ^8x \left( \csc ^4x-2\csc ^2x+1 \right)\cot x=\csc ^7x \left( \csc ^4x-2\csc ^2x+1 \right)\left( \csc x \cot x \right)$

2. anonymous

$=\left( \csc ^{11}x-2\csc ^9x+\csc ^7x \right)\left( \csc x \cot x \right)$

3. amurciano2

Thank you so much. How did you know to take out a cscxcotx and not csc^2x

4. anonymous

you can do that way also. $\csc ^6x \csc ^2x=(csc^2x)^3csc^2x=\left( \cot ^2x+1 \right)^3\csc ^2x$ $(\cot ^6x+3\cot ^4x+3\cot ^2x+1)\csc ^2x$

5. Error1603

g