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

The integral of cot^3(x/3) is 3(-(cot^2(x/3))/2-ln|sin(x/3)|)+C

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integral of cot^3(x/3)

\int cot^{3} (\frac{x}{3}) d x

3 (- \frac{cot ^{2} ( \frac{x}{3} )}{2} - ln sin (\frac{x}{3})) + C

Solution steps

\int cot^{3} (\frac{x}{3}) d x

Apply u-substitution

: \int cot^{3} (u) \cdot 3 d u

= \int cot^{3} (u) \cdot 3 d u

Take the constant out:

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

= 3 \cdot \int cot^{3} (u) d u

Apply Integral Reduction

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

\int cot (u) d u = ln ∣ sin (u) ∣

= 3 (- \frac{cot ^{2} ( u )}{2} - ln ∣ sin (u) ∣)

Substitute back

u = \frac{x}{3}

= 3 (- \frac{cot ^{2} ( \frac{x}{3} )}{2} - ln sin (\frac{x}{3}))

Add a constant to the solution

= 3 (- \frac{cot ^{2} ( \frac{x}{3} )}{2} - ln sin (\frac{x}{3})) + C

\int x ln^{4} (x) d x

\frac{\partial}{\partial x} (\frac{t}{c ^{2} t ^{2} - x ^{2}})

f (x) = \frac{12}{x + 7}

\frac{\partial}{\partial x} (\frac{2 x - y}{5 x y + 6})

f (t) = arcsin (t)

What is the integral of cot^3(x/3) ?
The integral of cot^3(x/3) is 3(-(cot^2(x/3))/2-ln|sin(x/3)|)+C