What is the integral of 1/(csc^6(t)) ?

The integral of 1/(csc^6(t)) is -(cos(t)sin^5(t))/6+5/6 (-1/4 sin^3(t)cos(t)+3/8 (t-1/2 sin(2t)))+C

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integral of 1/(csc^6(t))

\int \frac{1}{csc ^{6} ( t )} d t

- \frac{cos ( t ) sin ^{5} ( t )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (t) cos (t) + \frac{3}{8} (t - \frac{1}{2} sin (2 t))) + C

Solution steps

\int \frac{1}{csc ^{6} ( t )} d t

Rewrite using trig identities

= \int sin^{6} (t) d t

Apply Integral Reduction

= - \frac{cos ( t ) sin ^{5} ( t )}{6} + \frac{5}{6} \cdot \int sin^{4} (t) d t

\int sin^{4} (t) d t = - \frac{1}{4} sin^{3} (t) cos (t) + \frac{3}{8} (t - \frac{1}{2} sin (2 t))

= - \frac{cos ( t ) sin ^{5} ( t )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (t) cos (t) + \frac{3}{8} (t - \frac{1}{2} sin (2 t)))

Add a constant to the solution

= - \frac{cos ( t ) sin ^{5} ( t )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (t) cos (t) + \frac{3}{8} (t - \frac{1}{2} sin (2 t))) + C

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What is the integral of 1/(csc^6(t)) ?
The integral of 1/(csc^6(t)) is -(cos(t)sin^5(t))/6+5/6 (-1/4 sin^3(t)cos(t)+3/8 (t-1/2 sin(2t)))+C