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

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

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integral of cot^6(2t)

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

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

Solution steps

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

Apply u-substitution

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

= \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

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

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

Simplify

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

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

Add a constant to the solution

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

\int 7 e^{- 0.5 x} d x

\frac{\partial}{\partial x} (\frac{1}{x y})

\int e^{3 - 4 x} d x

7 y^{^{''}} + 13 y^{^{'}} - 2 y = 0

x \to (\frac{π}{2}) + lim (6 e^{t a n (x)})

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