How do you find the integral of ${cot x}^{5} {csc x}^{2}$ $cotx^5cscx^2$ ?

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How do you find the integral of ${cot x}^{5} {csc x}^{2}$ $cotx^5cscx^2$ ?

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Answer 1 · 2015-03-05T20:35:55

Our required integral is,

$I = \int {cot}^{5} x {csc}^{2} x d x$ $I = intcot^5xcsc^2xdx$

First, we make the substitution,

$cot x = t$ $cotx=t$

If we differentiate both sides with respect to $t$ $t$ ,

$\frac{d}{d t} (cot x) = \frac{d t}{d t}$ $d/dt(cotx)=dt/dt$

Then we get,

$(- {csc}^{2} x) \frac{d x}{d t} = 1$ $(-csc^2x)dx/dt=1$

On rearraging,

${csc}^{2} x d x = - d t$ $csc^2xdx = -dt$

On substituting these values in our original integral, we get

$I = \int t^{5} d t$ $I = intt^5dt$

which can be solved trivially to get the solution

$\frac{t^{6}}{6} + C$ $t^6/6 + C$

On substituting the value of $t = cot x$ $t = cotx$ , we get,

$I = \frac{{cot}^{6} x}{6} + C$ $I = cot^6x/6+C$

In case your question was $\int {cot x}^{5} {csc x}^{2} d x$ $intcotx^5cscx^2dx$ the answer is far more complicated and beyond the scope of a straight-forward pen and paper solution. Maybe, the solution does not even exist.

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How do you find the integral of ${cot x}^{5} {csc x}^{2}$ $cotx^5cscx^2$ ?

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