Problems on Techniques of Integration

Use the Integration by Parts. Set

u &=& \arctan(2 x)\\
dv &=& dx\;.


du &=&\displaystyle \frac{2 }{4 x^2 + 1} dx\\
v &=& x\;.


\begin{displaymath}\int \arctan(2 x) dx = x \arctan(2 x) - \int x \frac{2}{4 x^2 + 1} dx = x \arctan(x) - \int \frac{2 x}{4 x^2 + 1}dx\end{displaymath}


\begin{displaymath}\int \arctan(2 x) dx = x \arctan(2 x) - \frac{1}{4}\ln(4 x^2+1) + C\;.\end{displaymath}

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