What is the integral of csc^4(ax) ?

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

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integral of csc^4(ax)

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

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

Solution steps

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

Apply u-substitution

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

Take the constant out:

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

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

Apply Integral Reduction

= \frac{1}{a} (- \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)

= \frac{1}{a} (- \frac{cos ( u ) csc ^{3} ( u )}{3} + \frac{2}{3} (- cot (u)))

Substitute back

u = a x

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

Simplify

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

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

Add a constant to the solution

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

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