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

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

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integral of cot^3(2t)

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

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

Solution steps

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

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 t

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

Add a constant to the solution

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

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