Question

Integrate cot^5(x) csc^8(x)

Answer 1

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

Answer 2

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

Answer 3

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

Answer 4

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

Answer 5

g