What is the integral of cot^5(3-2x)csc^3(3-2x) ?

The integral of cot^5(3-2x)csc^3(3-2x) is 1/14 csc^7(3-2x)-1/5 csc^5(3-2x)+1/6 csc^3(3-2x)+C

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integral of cot^5(3-2x)csc^3(3-2x)

\int cot^{5} (3 - 2 x) csc^{3} (3 - 2 x) d x

\frac{1}{14} csc^{7} (3 - 2 x) - \frac{1}{5} csc^{5} (3 - 2 x) + \frac{1}{6} csc^{3} (3 - 2 x) + C

Solution steps

\int cot^{5} (3 - 2 x) csc^{3} (3 - 2 x) d x

Apply u-substitution

: \int - \frac{1}{2} cot^{5} (u) csc^{3} (u) d u

= \int - \frac{1}{2} cot^{5} (u) csc^{3} (u) d u

Take the constant out:

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

= - \frac{1}{2} \cdot \int cot^{5} (u) csc^{3} (u) d u

Apply u-substitution

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

= - \frac{1}{2} \cdot \int - v^{2} (v^{2} - 1)^{2} d v

Take the constant out:

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

= - \frac{1}{2} (- \int v^{2} (v^{2} - 1)^{2} d v)

Expand

v^{2} (v^{2} - 1)^{2} : v^{6} - 2 v^{4} + v^{2}

= - \frac{1}{2} (- \int v^{6} - 2 v^{4} + v^{2} d v)

Apply the Sum Rule:

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

= - \frac{1}{2} (- (\int v^{6} d v - \int 2 v^{4} d v + \int v^{2} d v))

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

\int 2 v^{4} d v = \frac{2 v ^{5}}{5}

\int v^{2} d v = \frac{v ^{3}}{3}

= - \frac{1}{2} (- (\frac{v ^{7}}{7} - \frac{2 v ^{5}}{5} + \frac{v ^{3}}{3}))

Substitute back

= - \frac{1}{2} (- (\frac{csc ^{7} ( 3 - 2 x )}{7} - \frac{2 csc ^{5} ( 3 - 2 x )}{5} + \frac{csc ^{3} ( 3 - 2 x )}{3}))

Simplify

- \frac{1}{2} (- (\frac{csc ^{7} ( 3 - 2 x )}{7} - \frac{2 csc ^{5} ( 3 - 2 x )}{5} + \frac{csc ^{3} ( 3 - 2 x )}{3})) : \frac{1}{14} csc^{7} (3 - 2 x) - \frac{1}{5} csc^{5} (3 - 2 x) + \frac{1}{6} csc^{3} (3 - 2 x)

= \frac{1}{14} csc^{7} (3 - 2 x) - \frac{1}{5} csc^{5} (3 - 2 x) + \frac{1}{6} csc^{3} (3 - 2 x)

Add a constant to the solution

= \frac{1}{14} csc^{7} (3 - 2 x) - \frac{1}{5} csc^{5} (3 - 2 x) + \frac{1}{6} csc^{3} (3 - 2 x) + C

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What is the integral of cot^5(3-2x)csc^3(3-2x) ?
The integral of cot^5(3-2x)csc^3(3-2x) is 1/14 csc^7(3-2x)-1/5 csc^5(3-2x)+1/6 csc^3(3-2x)+C