I would like to evaluate $$\int \dfrac{3x^2-1}{2x\sqrt{x}}\arctan(x){\rm d}x$$ I'm not even sure how to start, any suggestion will be appreciated.
2026-04-07 20:31:52.1775593912
Evaluating this integral $\int \frac{3x^2-1}{2x\sqrt{x}}\arctan(x){\rm d}x$, how to start?
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Hint. Integration by parts is a nice route, $$ \begin{align} \int \dfrac{3x^2-1}{2x\sqrt{x}}\arctan(x){\rm d}x&=\int \left(\frac{3\sqrt{x}}{2}-\frac1{2x^{3/2}} \right)\arctan(x){\rm d}x \\\\&=\left(x^{3/2}+\frac{1}{\sqrt{x}}\right)\arctan(x)-\int \left(\frac{x^2+1}{\sqrt{x}} \right)\left(\arctan(x)\right)'{\rm d}x \end{align} $$ Can you finish it?