What is the integral of csc^4(x/4) ?

The integral of csc^4(x/4) is 4(-1/3 csc^3(x/4)cos(x/4)-2/3 cot(x/4))+C

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integral of csc^4(x/4)

\int csc^{4} (\frac{x}{4}) d x

4 (- \frac{1}{3} csc^{3} (\frac{x}{4}) cos (\frac{x}{4}) - \frac{2}{3} cot (\frac{x}{4})) + C

Solution steps

\int csc^{4} (\frac{x}{4}) d x

Apply u-substitution

= \int csc^{4} (u) \cdot 4 d u

Take the constant out:

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

= 4 \cdot \int csc^{4} (u) d u

Apply Integral Reduction

= 4 (- \frac{cos ( u ) csc ^{3} ( u )}{3} + \frac{2}{3} \cdot \int csc^{2} (u) d u)

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

= 4 (- \frac{cos ( u ) csc ^{3} ( u )}{3} + \frac{2}{3} (- cot (u)))

Substitute back

u = \frac{x}{4}

= 4 (- \frac{cos ( \frac{x}{4} ) csc ^{3} ( \frac{x}{4} )}{3} + \frac{2}{3} (- cot (\frac{x}{4})))

Simplify

4 (- \frac{cos ( \frac{x}{4} ) csc ^{3} ( \frac{x}{4} )}{3} + \frac{2}{3} (- cot (\frac{x}{4}))) : 4 (- \frac{1}{3} csc^{3} (\frac{x}{4}) cos (\frac{x}{4}) - \frac{2}{3} cot (\frac{x}{4}))

= 4 (- \frac{1}{3} csc^{3} (\frac{x}{4}) cos (\frac{x}{4}) - \frac{2}{3} cot (\frac{x}{4}))

Add a constant to the solution

= 4 (- \frac{1}{3} csc^{3} (\frac{x}{4}) cos (\frac{x}{4}) - \frac{2}{3} cot (\frac{x}{4})) + C

What is the integral of csc^4(x/4) ?
The integral of csc^4(x/4) is 4(-1/3 csc^3(x/4)cos(x/4)-2/3 cot(x/4))+C