What is the integral of cot^3(n)csc^5(n) ?

The integral of cot^3(n)csc^5(n) is -1/7 csc^7(n)+1/5 csc^5(n)+C

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integral of cot^3(n)csc^5(n)

\int cot^{3} (n) csc^{5} (n) d n

- \frac{1}{7} csc^{7} (n) + \frac{1}{5} csc^{5} (n) + C

Solution steps

\int cot^{3} (n) csc^{5} (n) d n

Apply u-substitution

: \int - u^{4} (u^{2} - 1) d u

= \int - u^{4} (u^{2} - 1) d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int u^{4} (u^{2} - 1) d u

Expand

u^{4} (u^{2} - 1) : u^{6} - u^{4}

= - \int u^{6} - u^{4} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - (\int u^{6} d u - \int u^{4} d u)

\int u^{6} d u = \frac{u ^{7}}{7}

\int u^{4} d u = \frac{u ^{5}}{5}

= - (\frac{u ^{7}}{7} - \frac{u ^{5}}{5})

Substitute back

u = csc (n)

= - (\frac{csc ^{7} ( n )}{7} - \frac{csc ^{5} ( n )}{5})

Simplify

- (\frac{csc ^{7} ( n )}{7} - \frac{csc ^{5} ( n )}{5}) : - \frac{1}{7} csc^{7} (n) + \frac{1}{5} csc^{5} (n)

= - \frac{1}{7} csc^{7} (n) + \frac{1}{5} csc^{5} (n)

Add a constant to the solution

= - \frac{1}{7} csc^{7} (n) + \frac{1}{5} csc^{5} (n) + C

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What is the integral of cot^3(n)csc^5(n) ?
The integral of cot^3(n)csc^5(n) is -1/7 csc^7(n)+1/5 csc^5(n)+C