What is the integral of 2csc^2(x/3) ?

The integral of 2csc^2(x/3) is -6cot(x/3)+C

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integral of 2csc^2(x/3)

\int 2 csc^{2} (\frac{x}{3}) d x

- 6 cot (\frac{x}{3}) + C

Solution steps

\int 2 csc^{2} (\frac{x}{3}) d x

Take the constant out:

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

= 2 \cdot \int csc^{2} (\frac{x}{3}) d x

Rewrite using trig identities

= 2 \cdot \int 1 + cot^{2} (\frac{x}{3}) 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

= 2 (\int 1 d x + \int cot^{2} (\frac{x}{3}) d x)

\int 1 d x = x

\int cot^{2} (\frac{x}{3}) d x = - x - 3 cot (\frac{x}{3})

= 2 (x - x - 3 cot (\frac{x}{3}))

Simplify

= - 6 cot (\frac{x}{3})

Add a constant to the solution

= - 6 cot (\frac{x}{3}) + C

What is the integral of 2csc^2(x/3) ?
The integral of 2csc^2(x/3) is -6cot(x/3)+C