What is the antiderivative of $\csc^2 \sqrt{x}$?

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Please how can I solve the antiderivative of $\csc^2 \sqrt{x}$ ?

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1 Answer

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$$\begin{align}\int dx \, \csc^2{\sqrt{x}} &= 2 \int du \, u \, \csc^2{u} \\ &= -2u \, \cot{u} + 2\int du \, \cot{u} \\ &= -2u \, \cot{u} + 2\log{|\sin{u}|} + C \\ &= -2\sqrt{x} \, \cot{\sqrt{x}} + \log{\sin^2{\sqrt{x}}} + C \end{align} $$

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