Solve ∫ cot(2/3x)*{left(sec(2/3x)right)}^2 wrt x

$$\int{ \cot ( \frac{ 2 }{ 3 } x ) \times { \left( \sec ( \frac{ 2 }{ 3 } x ) \right) }^{ 2 } }d x$$

Evaluate

$$\frac{3\ln(|\sin(\frac{2x}{3})|)}{2}-\frac{3\ln(|\cos(\frac{2x}{3})|)}{2}+С$$

Solution Steps

Simplify.

$$\frac{-3\ln(|\cos(\frac{2x}{3})|)+3\ln(|\sin(\frac{2x}{3})|)}{2}$$

If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$-\frac{3\ln(|\cos(\frac{2x}{3})|)}{2}+\frac{3\ln(|\sin(\frac{2x}{3})|)}{2}+С$$

Differentiate w.r.t. x

$$\frac{2}{\sin(\frac{4x}{3})}$$