csc cot Power Integral Problem

[Jason76]

\(\displaystyle \int 8 csc^{4}(x) \cot^{6}(x) dx\)

\(\displaystyle \int 8[(csc^{2}(x))(\csc^{2}(x)) \cot^{6}(x) dx\)

\(\displaystyle \int 8[(\cot^{2}{x} + 1)(\csc^{2}(x))(\cot^{6}(x)) dx\)

\(\displaystyle \int 8[(\cot^{2}(x) + 1)(\csc^{2}(x))(\cot^{6}(x)) dx\)

\(\displaystyle \int 8[(\cot^{2}(x) + 1))(\cot^{6}(x) (\csc^{2}(x)) dx\) - rearranged to show derivative of cot on right

\(\displaystyle u = \cot x\)

\(\displaystyle du = -\csc^{2}(x)\)

\(\displaystyle -du = \csc^{2}(x)\)

\(\displaystyle -\int 8[(u^{2} + 1)(u^{6}) du\)

\(\displaystyle -\int 8[(u^{12} + u^{6})]\)

\(\displaystyle = -8\dfrac{u^{13}}{13} + \dfrac{u^{7}}{7} + C\)

\(\displaystyle = -8\dfrac{cot^{13}(x)}{13} + \dfrac{\cot^{7}(x)}{7} + C\) ??

 

[Deleted member 4993]

\(\displaystyle \int 8 csc^{4}(x) \cot^{6}(x) dx\)

\(\displaystyle \int 8[(csc^{2}(x))(\csc^{2}(x)) \cot^{6}(x) dx\)

\(\displaystyle \int 8[(\cot^{2}{x} + 1)(\csc^{2}(x))(\cot^{6}(x)) dx\)

\(\displaystyle \int 8[(\cot^{2}(x) + 1)(\csc^{2}(x))(\cot^{6}(x)) dx\)

\(\displaystyle \int 8[(\cot^{2}(x) + 1))(\cot^{6}(x) (\csc^{2}(x)) dx\) - rearranged to show derivative of cot on right

\(\displaystyle u = \cot x\)

\(\displaystyle du = -\csc^{2}(x)\)

\(\displaystyle -du = \csc^{2}(x)\)

\(\displaystyle -\int 8[(u^{2} + 1)(u^{6}) du\)

\(\displaystyle -\int 8[(u^{12} + u^{6})]du\) ...incorrect should be \(\displaystyle -\int 8[(u^{6+2} + u^{6})]du \ = \ - \int 8[(u^{8} + u^{6})]du\)

Algebra is getting you again....

 

[Jason76]

\(\displaystyle \int 8 csc^{4}(x) \cot^{6}(x) dx\)

\(\displaystyle \int 8[(csc^{2}(x))(\csc^{2}(x)) \cot^{6}(x) dx\)

\(\displaystyle \int 8[(\cot^{2}{x} + 1)(\csc^{2}(x))(\cot^{6}(x)) dx\)

\(\displaystyle \int 8[(\cot^{2}(x) + 1)(\csc^{2}(x))(\cot^{6}(x)) dx\)

\(\displaystyle \int 8[(\cot^{2}(x) + 1))(\cot^{6}(x) (\csc^{2}(x)) dx\) - rearranged to show derivative of cot on right

\(\displaystyle u = \cot x\)

\(\displaystyle du = -\csc^{2}(x)\)

\(\displaystyle -du = \csc^{2}(x)\)

\(\displaystyle -\int 8[(u^{2} + 1)(u^{6})] du\)

\(\displaystyle -\int 8[(u^{8} + u^{6})]\)

\(\displaystyle -\int 8u^{8} + 8u^{6})\)

\(\displaystyle = -8\dfrac{u^{9}}{9} + 8\dfrac{u^{7}}{7} + C\)

\(\displaystyle = -8\dfrac{cot^{9}(x)}{9} + 8\dfrac{\cot^{7}(x)}{7} + C\)

 

[Jason76]

The negative sign should have been distributed along with the 8, as well an incorrect trig function in the final answer.