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

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

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

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

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

Solution steps

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

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 (x)

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

Simplify

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

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

Add a constant to the solution

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

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