What is the integral of cot^4(4x) ?

The integral of cot^4(4x) is -1/12 cot^3(4x)+x+1/4 cot(4x)+C

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integral of cot^4(4x)

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

- \frac{1}{12} cot^{3} (4 x) + x + \frac{1}{4} cot (4 x) + C

Solution steps

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

Apply u-substitution

: \int cot^{4} (u) \frac{1}{4} d u

= \int cot^{4} (u) \frac{1}{4} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int cot^{4} (u) d u

Apply Integral Reduction

= \frac{1}{4} (- \frac{cot ^{3} ( u )}{3} - \int cot^{2} (u) d u)

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

= \frac{1}{4} (- \frac{cot ^{3} ( u )}{3} - (- u - cot (u)))

Substitute back

u = 4 x

= \frac{1}{4} (- \frac{cot ^{3} ( 4 x )}{3} - (- 4 x - cot (4 x)))

Simplify

\frac{1}{4} (- \frac{cot ^{3} ( 4 x )}{3} - (- 4 x - cot (4 x))) : - \frac{1}{12} cot^{3} (4 x) + x + \frac{1}{4} cot (4 x)

= - \frac{1}{12} cot^{3} (4 x) + x + \frac{1}{4} cot (4 x)

Add a constant to the solution

= - \frac{1}{12} cot^{3} (4 x) + x + \frac{1}{4} cot (4 x) + C

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What is the integral of cot^4(4x) ?
The integral of cot^4(4x) is -1/12 cot^3(4x)+x+1/4 cot(4x)+C