\(\displaystyle \int 8 csc^{4}(x) \cot^{6}(x) dx\)
\(\displaystyle \int 8[(csc^{2}(x))(\csc^{2}(x)) \cot^{6}(x) dx\)
\(\displaystyle \int 8[(\cot^{2}{x} + 1)(\csc^{2}(x))(\cot^{6}(x)) dx\)
\(\displaystyle \int 8[(\cot^{2}(x) + 1)(\csc^{2}(x))(\cot^{6}(x)) dx\)
\(\displaystyle \int 8[(\cot^{2}(x) + 1))(\cot^{6}(x) (\csc^{2}(x)) dx\) - rearranged to show derivative of cot on right
\(\displaystyle u = \cot x\)
\(\displaystyle du = -\csc^{2}(x)\)
\(\displaystyle -du = \csc^{2}(x)\)
\(\displaystyle -\int 8[(u^{2} + 1)(u^{6}) du\)
\(\displaystyle -\int 8[(u^{12} + u^{6})]\)
\(\displaystyle = -8\dfrac{u^{13}}{13} + \dfrac{u^{7}}{7} + C\)
\(\displaystyle = -8\dfrac{cot^{13}(x)}{13} + \dfrac{\cot^{7}(x)}{7} + C\) ??
\(\displaystyle \int 8[(csc^{2}(x))(\csc^{2}(x)) \cot^{6}(x) dx\)
\(\displaystyle \int 8[(\cot^{2}{x} + 1)(\csc^{2}(x))(\cot^{6}(x)) dx\)
\(\displaystyle \int 8[(\cot^{2}(x) + 1)(\csc^{2}(x))(\cot^{6}(x)) dx\)
\(\displaystyle \int 8[(\cot^{2}(x) + 1))(\cot^{6}(x) (\csc^{2}(x)) dx\) - rearranged to show derivative of cot on right
\(\displaystyle u = \cot x\)
\(\displaystyle du = -\csc^{2}(x)\)
\(\displaystyle -du = \csc^{2}(x)\)
\(\displaystyle -\int 8[(u^{2} + 1)(u^{6}) du\)
\(\displaystyle -\int 8[(u^{12} + u^{6})]\)
\(\displaystyle = -8\dfrac{u^{13}}{13} + \dfrac{u^{7}}{7} + C\)
\(\displaystyle = -8\dfrac{cot^{13}(x)}{13} + \dfrac{\cot^{7}(x)}{7} + C\) ??
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