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

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

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integral of (csc(3x)-cot(3x))^2

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

- x - \frac{2}{3} cot (3 x) + \frac{2}{3} csc (3 x) + C

Solution steps

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

Expand

(csc (3 x) - cot (3 x))^{2} : csc^{2} (3 x) - 2 csc (3 x) cot (3 x) + cot^{2} (3 x)

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

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

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

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

= - \frac{1}{3} cot (3 x) - (- \frac{2}{3} csc (3 x)) - x - \frac{1}{3} cot (3 x)

Simplify

- \frac{1}{3} cot (3 x) - (- \frac{2}{3} csc (3 x)) - x - \frac{1}{3} cot (3 x) : - x - \frac{2}{3} cot (3 x) + \frac{2}{3} csc (3 x)

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

Add a constant to the solution

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

y^{^{''}} - 2 y^{^{'}} = sin (x), y (0) = 1, y^{^{'}} (0) = 2

\int t^{2} + 4 d t

\frac{d}{d x} (sin (x) + \frac{5}{6} cot (x))

\int_{- 8}^{0} (4 - 9 x)^{- 0.5} d x

(x - 1)^{3}, (x - 1)

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