What is the integral of cot^5(6x)csc^7(6x) ?

The integral of cot^5(6x)csc^7(6x) is 1/6 (-1/11 csc^{11}(6x)+2/9 csc^9(6x)-1/7 csc^7(6x))+C

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integral of cot^5(6x)csc^7(6x)

\int cot^{5} (6 x) csc^{7} (6 x) d x

\frac{1}{6} (- \frac{1}{11} csc^{11} (6 x) + \frac{2}{9} csc^{9} (6 x) - \frac{1}{7} csc^{7} (6 x)) + C

Solution steps

\int cot^{5} (6 x) csc^{7} (6 x) d x

Apply u-substitution

: \int cot^{5} (u) csc^{7} (u) \frac{1}{6} d u

= \int cot^{5} (u) csc^{7} (u) \frac{1}{6} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int cot^{5} (u) csc^{7} (u) d u

Rewrite using trig identities

= \frac{1}{6} \cdot \int \frac{( 1 - sin ^{2} ( u ) ) ^{2} cos ( u )}{sin ^{12} ( u )} d u

Apply u-substitution

: \int \frac{( 1 - v ^{2} ) ^{2}}{v ^{12}} d v

= \frac{1}{6} \cdot \int \frac{( 1 - v ^{2} ) ^{2}}{v ^{12}} d v

Expand

\frac{( 1 - v ^{2} ) ^{2}}{v ^{12}} : \frac{1}{v ^{12}} - \frac{2}{v ^{10}} + \frac{1}{v ^{8}}

= \frac{1}{6} \cdot \int \frac{1}{v ^{12}} - \frac{2}{v ^{10}} + \frac{1}{v ^{8}} 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}{6} (\int \frac{1}{v ^{12}} d v - \int \frac{2}{v ^{10}} d v + \int \frac{1}{v ^{8}} d v)

\int \frac{1}{v ^{12}} d v = - \frac{1}{11 v ^{11}}

\int \frac{2}{v ^{10}} d v = - \frac{2}{9 v ^{9}}

\int \frac{1}{v ^{8}} d v = - \frac{1}{7 v ^{7}}

= \frac{1}{6} (- \frac{1}{11 v ^{11}} - (- \frac{2}{9 v ^{9}}) - \frac{1}{7 v ^{7}})

Substitute back

= \frac{1}{6} (- \frac{1}{11 sin ^{11} ( 6 x )} - (- \frac{2}{9 sin ^{9} ( 6 x )}) - \frac{1}{7 sin ^{7} ( 6 x )})

Simplify

\frac{1}{6} (- \frac{1}{11 sin ^{11} ( 6 x )} - (- \frac{2}{9 sin ^{9} ( 6 x )}) - \frac{1}{7 sin ^{7} ( 6 x )}) : \frac{1}{6} (- \frac{1}{11} csc^{11} (6 x) + \frac{2}{9} csc^{9} (6 x) - \frac{1}{7} csc^{7} (6 x))

= \frac{1}{6} (- \frac{1}{11} csc^{11} (6 x) + \frac{2}{9} csc^{9} (6 x) - \frac{1}{7} csc^{7} (6 x))

Add a constant to the solution

= \frac{1}{6} (- \frac{1}{11} csc^{11} (6 x) + \frac{2}{9} csc^{9} (6 x) - \frac{1}{7} csc^{7} (6 x)) + C

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What is the integral of cot^5(6x)csc^7(6x) ?
The integral of cot^5(6x)csc^7(6x) is 1/6 (-1/11 csc^{11}(6x)+2/9 csc^9(6x)-1/7 csc^7(6x))+C