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

The integral of cot^3(2x) is 1/2 (-(cot^2(2x))/2-ln|sin(2x)|)+C

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integral of cot^3(2x)

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

\frac{1}{2} (- \frac{cot ^{2} ( 2 x )}{2} - ln ∣ sin (2 x) ∣) + C

Solution steps

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

Apply u-substitution

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

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

Apply Integral Reduction

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

\int cot (u) d u = ln ∣ sin (u) ∣

= \frac{1}{2} (- \frac{cot ^{2} ( u )}{2} - ln ∣ sin (u) ∣)

Substitute back

u = 2 x

= \frac{1}{2} (- \frac{cot ^{2} ( 2 x )}{2} - ln ∣ sin (2 x) ∣)

Add a constant to the solution

= \frac{1}{2} (- \frac{cot ^{2} ( 2 x )}{2} - ln ∣ sin (2 x) ∣) + C

(x^{2} + 4)^{7}

x \to a lim (2 - x)

y^{^{''}} - 6 y^{^{'}} + 9 y = 24 - 9 t - (12 t e^{3 t} + 4 e^{3 t})

x \to 4 - lim (x)

d x + (x + y + 1) d y = 0

What is the integral of cot^3(2x) ?
The integral of cot^3(2x) is 1/2 (-(cot^2(2x))/2-ln|sin(2x)|)+C