What is the integral of csc^8(x) ?

The integral of csc^8(x) is -1/7 csc^7(x)cos(x)+2/35 (4(-csc^3(x)cos(x)-2cot(x))-3csc^5(x)cos(x))+C

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integral of csc^8(x)

\int csc^{8} (x) d x

- \frac{1}{7} csc^{7} (x) cos (x) + \frac{2}{35} (4 (- csc^{3} (x) cos (x) - 2 cot (x)) - 3 csc^{5} (x) cos (x)) + C

Solution steps

\int csc^{8} (x) d x

Apply Integral Reduction

= - \frac{cos ( x ) csc ^{7} ( x )}{7} + \frac{6}{7} \cdot \int csc^{6} (x) d x

\int csc^{6} (x) d x = - \frac{cos ( x ) csc ^{5} ( x )}{5} + \frac{4}{5} (- \frac{1}{3} csc^{3} (x) cos (x) - \frac{2}{3} cot (x))

= - \frac{cos ( x ) csc ^{7} ( x )}{7} + \frac{6}{7} (- \frac{cos ( x ) csc ^{5} ( x )}{5} + \frac{4}{5} (- \frac{1}{3} csc^{3} (x) cos (x) - \frac{2}{3} cot (x)))

Simplify

- \frac{cos ( x ) csc ^{7} ( x )}{7} + \frac{6}{7} (- \frac{cos ( x ) csc ^{5} ( x )}{5} + \frac{4}{5} (- \frac{1}{3} csc^{3} (x) cos (x) - \frac{2}{3} cot (x))) : - \frac{1}{7} csc^{7} (x) cos (x) + \frac{2}{35} (4 (- csc^{3} (x) cos (x) - 2 cot (x)) - 3 csc^{5} (x) cos (x))

= - \frac{1}{7} csc^{7} (x) cos (x) + \frac{2}{35} (4 (- csc^{3} (x) cos (x) - 2 cot (x)) - 3 csc^{5} (x) cos (x))

Add a constant to the solution

= - \frac{1}{7} csc^{7} (x) cos (x) + \frac{2}{35} (4 (- csc^{3} (x) cos (x) - 2 cot (x)) - 3 csc^{5} (x) cos (x)) + C

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\frac{\partial}{\partial x} (lo g_{3} (7^{x^{2} - 4}))

What is the integral of csc^8(x) ?
The integral of csc^8(x) is -1/7 csc^7(x)cos(x)+2/35 (4(-csc^3(x)cos(x)-2cot(x))-3csc^5(x)cos(x))+C