What is the integral of cot^6(2x) ?

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

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integral of cot^6(2x)

\int cot^{6} (2 x) d x

- \frac{1}{2} (\frac{1}{5} cot^{5} (2 x) - \frac{1}{3} cot^{3} (2 x) + cot (2 x) - arctan (cot (2 x))) + C

Solution steps

\int cot^{6} (2 x) d x

Apply u-substitution

= \int cot^{6} (u) \frac{1}{2} 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^{6} (u) d u

Apply u-substitution

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

Long division

\frac{v ^{6}}{1 + v ^{2}} : v^{4} - v^{2} + 1 - \frac{1}{v ^{2} + 1}

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

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

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

\int 1 d v = v

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

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

Substitute back

= \frac{1}{2} (- (\frac{cot ^{5} ( 2 x )}{5} - \frac{cot ^{3} ( 2 x )}{3} + cot (2 x) - arctan (cot (2 x))))

Simplify

\frac{1}{2} (- (\frac{cot ^{5} ( 2 x )}{5} - \frac{cot ^{3} ( 2 x )}{3} + cot (2 x) - arctan (cot (2 x)))) : - \frac{1}{2} (\frac{1}{5} cot^{5} (2 x) - \frac{1}{3} cot^{3} (2 x) + cot (2 x) - arctan (cot (2 x)))

= - \frac{1}{2} (\frac{1}{5} cot^{5} (2 x) - \frac{1}{3} cot^{3} (2 x) + cot (2 x) - arctan (cot (2 x)))

Add a constant to the solution

= - \frac{1}{2} (\frac{1}{5} cot^{5} (2 x) - \frac{1}{3} cot^{3} (2 x) + cot (2 x) - arctan (cot (2 x))) + C

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