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Popular >

integral of csc^{15}(x)

Solution

\int csc^{15} (x) d x

Solution

- \frac{1}{14} csc^{14} (x) cos (x) + \frac{13}{215040} (- 1280 csc^{12} (x) cos (x) + 11 (- 128 csc^{10} (x) cos (x) + 3 (7 (5 (3 (ln tan (\frac{x}{2}) - cot (x) csc (x)) - 2 csc^{4} (x) cos (x)) - 8 csc^{6} (x) cos (x)) - 48 csc^{8} (x) cos (x)))) + C

Solution steps

\int csc^{15} (x) d x

Apply Integral Reduction

= - \frac{cos ( x ) csc ^{14} ( x )}{14} + \frac{13}{14} \cdot \int csc^{13} (x) d x

\int csc^{13} (x) d x = - \frac{cos ( x ) csc ^{12} ( x )}{12} + \frac{11}{12} (- \frac{1}{10} csc^{10} (x) cos (x) + \frac{3}{1280} (7 (5 (3 (ln tan (\frac{x}{2}) - cot (x) csc (x)) - 2 csc^{4} (x) cos (x)) - 8 csc^{6} (x) cos (x)) - 48 csc^{8} (x) cos (x)))

= - \frac{cos ( x ) csc ^{14} ( x )}{14} + \frac{13}{14} (- \frac{cos ( x ) csc ^{12} ( x )}{12} + \frac{11}{12} (- \frac{1}{10} csc^{10} (x) cos (x) + \frac{3}{1280} (7 (5 (3 (ln tan (\frac{x}{2}) - cot (x) csc (x)) - 2 csc^{4} (x) cos (x)) - 8 csc^{6} (x) cos (x)) - 48 csc^{8} (x) cos (x))))

Simplify

- \frac{cos ( x ) csc ^{14} ( x )}{14} + \frac{13}{14} (- \frac{cos ( x ) csc ^{12} ( x )}{12} + \frac{11}{12} (- \frac{1}{10} csc^{10} (x) cos (x) + \frac{3}{1280} (7 (5 (3 (ln tan (\frac{x}{2}) - cot (x) csc (x)) - 2 csc^{4} (x) cos (x)) - 8 csc^{6} (x) cos (x)) - 48 csc^{8} (x) cos (x)))) : - \frac{1}{14} csc^{14} (x) cos (x) + \frac{13}{215040} (- 1280 csc^{12} (x) cos (x) + 11 (- 128 csc^{10} (x) cos (x) + 3 (7 (5 (3 (ln tan (\frac{x}{2}) - cot (x) csc (x)) - 2 csc^{4} (x) cos (x)) - 8 csc^{6} (x) cos (x)) - 48 csc^{8} (x) cos (x))))

= - \frac{1}{14} csc^{14} (x) cos (x) + \frac{13}{215040} (- 1280 csc^{12} (x) cos (x) + 11 (- 128 csc^{10} (x) cos (x) + 3 (7 (5 (3 (ln tan (\frac{x}{2}) - cot (x) csc (x)) - 2 csc^{4} (x) cos (x)) - 8 csc^{6} (x) cos (x)) - 48 csc^{8} (x) cos (x))))

Add a constant to the solution

= - \frac{1}{14} csc^{14} (x) cos (x) + \frac{13}{215040} (- 1280 csc^{12} (x) cos (x) + 11 (- 128 csc^{10} (x) cos (x) + 3 (7 (5 (3 (ln tan (\frac{x}{2}) - cot (x) csc (x)) - 2 csc^{4} (x) cos (x)) - 8 csc^{6} (x) cos (x)) - 48 csc^{8} (x) cos (x)))) + C

Graph

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Frequently Asked Questions (FAQ)

What is the integral of csc^{15}(x) ?
The integral of csc^{15}(x) is