What is the integral of csc^2(x/5) ?

The integral of csc^2(x/5) is -5cot(x/5)+C

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integral of csc^2(x/5)

\int csc^{2} (\frac{x}{5}) d x

- 5 cot (\frac{x}{5}) + C

Solution steps

\int csc^{2} (\frac{x}{5}) d x

Rewrite using trig identities

= \int 1 + cot^{2} (\frac{x}{5}) 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}{5}) d x

\int 1 d x = x

\int cot^{2} (\frac{x}{5}) d x = - x - 5 cot (\frac{x}{5})

= x - x - 5 cot (\frac{x}{5})

Simplify

= - 5 cot (\frac{x}{5})

Add a constant to the solution

= - 5 cot (\frac{x}{5}) + C

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