What is the integral of csc^2(x/8) ?

The integral of csc^2(x/8) is -8cot(x/8)+C

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integral of csc^2(x/8)

\int csc^{2} (\frac{x}{8}) d x

- 8 cot (\frac{x}{8}) + C

Solution steps

\int csc^{2} (\frac{x}{8}) d x

Rewrite using trig identities

= \int 1 + cot^{2} (\frac{x}{8}) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 d x + \int cot^{2} (\frac{x}{8}) d x

\int 1 d x = x

\int cot^{2} (\frac{x}{8}) d x = - x - 8 cot (\frac{x}{8})

= x - x - 8 cot (\frac{x}{8})

Simplify

= - 8 cot (\frac{x}{8})

Add a constant to the solution

= - 8 cot (\frac{x}{8}) + C

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- \frac{x ^{4}}{y ^{4}}