Obtain a formula for \(\displaystyle\displaystyle\int\limits_{\frac{1}{2}}^{n}{\frac{{{\left( {

Question

Obtain a formula for $\displaystyle\displaystyle\int\limits_{\frac{1}{2}}^{n}{\frac{{{\left( {{\tan }^{-1}}2x \right)}^{2}}}{1+4{{x}^{2}}}}\mathrm{ d}x$ in terms of $n$, where $n>0$. Hence evaluate $\displaystyle\displaystyle\int\limits_{\frac{1}{2}}^{\infty }{\frac{{{\left( {{\tan }^{-1}}2x \right)}^{2}}}{1+4{{x}^{2}}}}\mathrm{ d}x$ exactly.[4]

Accepted Answer

$\frac{1}{6}\left[ {{\left( \frac{\pi }{2} \right)}^{3}}-{{\left( \frac{\pi }{4} \right)}^{3}} \right]$; $\frac{7}{384}{{\pi }^{3}}$

2012 DHS P1 Q2

2012 DHS P1 Q2

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