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

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

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integral of csc^2(x/3)

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

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

Solution steps

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

Rewrite using trig identities

= \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

= \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})

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

Simplify

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

Add a constant to the solution

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

\frac{\partial}{\partial y} (5 y^{2} e^{5 x y})

\int sin (wt) d t

x y^{^{'}} = \frac{1}{y ^{3}}

\int_{- 1}^{2} (- x^{2} + x + 2)^{2} d x

x \to \infty lim (\frac{x}{0})