What is the integral of e^xcot^8(e^x) ?

The integral of e^xcot^8(e^x) is -1/7 cot^7(e^x)+1/5 cot^5(e^x)-1/3 cot^3(e^x)+e^x+cot(e^x)+C

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integral of e^xcot^8(e^x)

\int e^{x} cot^{8} (e^{x}) d x

- \frac{1}{7} cot^{7} (e^{x}) + \frac{1}{5} cot^{5} (e^{x}) - \frac{1}{3} cot^{3} (e^{x}) + e^{x} + cot (e^{x}) + C

Solution steps

\int e^{x} cot^{8} (e^{x}) d x

Apply u-substitution

: \int cot^{8} (u) d u

= \int cot^{8} (u) d u

Apply Integral Reduction

= - \frac{cot ^{7} ( u )}{7} - \int cot^{6} (u) d u

\int cot^{6} (u) d u = - \frac{1}{5} cot^{5} (u) + \frac{1}{3} cot^{3} (u) - u - cot (u)

= - \frac{cot ^{7} ( u )}{7} - (- \frac{1}{5} cot^{5} (u) + \frac{1}{3} cot^{3} (u) - u - cot (u))

Substitute back

u = e^{x}

= - \frac{cot ^{7} ( e ^{x} )}{7} - (- \frac{1}{5} cot^{5} (e^{x}) + \frac{1}{3} cot^{3} (e^{x}) - e^{x} - cot (e^{x}))

Simplify

- \frac{cot ^{7} ( e ^{x} )}{7} - (- \frac{1}{5} cot^{5} (e^{x}) + \frac{1}{3} cot^{3} (e^{x}) - e^{x} - cot (e^{x})) : - \frac{1}{7} cot^{7} (e^{x}) + \frac{1}{5} cot^{5} (e^{x}) - \frac{1}{3} cot^{3} (e^{x}) + e^{x} + cot (e^{x})

= - \frac{1}{7} cot^{7} (e^{x}) + \frac{1}{5} cot^{5} (e^{x}) - \frac{1}{3} cot^{3} (e^{x}) + e^{x} + cot (e^{x})

Add a constant to the solution

= - \frac{1}{7} cot^{7} (e^{x}) + \frac{1}{5} cot^{5} (e^{x}) - \frac{1}{3} cot^{3} (e^{x}) + e^{x} + cot (e^{x}) + C

\int \frac{sec ( ln ( x ) )}{x} d x

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\int sin (x) cot^{2} (x) d x

\int a x^{- 1} d x

\int \frac{1}{( x + 1 ) ( x + 3 ) ^{2}} d x

What is the integral of e^xcot^8(e^x) ?
The integral of e^xcot^8(e^x) is -1/7 cot^7(e^x)+1/5 cot^5(e^x)-1/3 cot^3(e^x)+e^x+cot(e^x)+C