What is the integral of 8/9 csc^2(x/9) ?

The integral of 8/9 csc^2(x/9) is -8cot(x/9)+C

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integral of 8/9 csc^2(x/9)

\int \frac{8}{9} csc^{2} (\frac{x}{9}) d x

- 8 cot (\frac{x}{9}) + C

Solution steps

\int \frac{8}{9} csc^{2} (\frac{x}{9}) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{8}{9} \cdot \int csc^{2} (\frac{x}{9}) d x

Rewrite using trig identities

= \frac{8}{9} \cdot \int 1 + cot^{2} (\frac{x}{9}) 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

= \frac{8}{9} (\int 1 d x + \int cot^{2} (\frac{x}{9}) d x)

\int 1 d x = x

\int cot^{2} (\frac{x}{9}) d x = - x - 9 cot (\frac{x}{9})

= \frac{8}{9} (x - x - 9 cot (\frac{x}{9}))

Simplify

\frac{8}{9} (x - x - 9 cot (\frac{x}{9})) : - 8 cot (\frac{x}{9})

= - 8 cot (\frac{x}{9})

Add a constant to the solution

= - 8 cot (\frac{x}{9}) + C

What is the integral of 8/9 csc^2(x/9) ?
The integral of 8/9 csc^2(x/9) is -8cot(x/9)+C