$\newcommand{\+}{^{\dagger}}% \newcommand{\angles}[1]{\left\langle #1 \right\rangle}% \newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}% \newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}% \newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}% \newcommand{\dd}{{\rm d}}% \newcommand{\down}{\downarrow}% \newcommand{\ds}[1]{\displaystyle{#1}}% \newcommand{\equalby}[1]{{#1 \atop {= \atop \vphantom{\huge A}}}}% \newcommand{\expo}[1]{\,{\rm e}^{#1}\,}% \newcommand{\fermi}{\,{\rm f}}% \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,}% \newcommand{\half}{{1 \over 2}}% \newcommand{\ic}{{\rm i}}% \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow}% \newcommand{\isdiv}{\,\left.\right\vert\,}% \newcommand{\ket}[1]{\left\vert #1\right\rangle}% \newcommand{\ol}[1]{\overline{#1}}% \newcommand{\pars}[1]{\left( #1 \right)}% \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}}% \newcommand{\root}[2][]{\,\sqrt[#1]{\,#2\,}\,}% \newcommand{\sech}{\,{\rm sech}}% \newcommand{\sgn}{\,{\rm sgn}}% \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}}% \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$ \begin{align} I_{n} \equiv \int_{1}^{n}{\dd x \over \pars{x^{2} + 1}^{n}}&= {n \over \pars{n^{2} + 1}^{n}} - {1 \over 2^{n}} + 2n\int_{1}^{n}{x^{2} \over \pars{x^{2} + 1}^{n + 1}}\,\dd x \\[3mm]&= {n \over \pars{n^{2} + 1}^{n}} - {1 \over 2^{n}} + 2nI_{n} - 2n\int_{1}^{n}{\dd x \over \pars{x^{2} + 1}^{n + 1}} \end{align}
\begin{align} I_{n} &= {n \over \pars{1 - 2n}\pars{n^{2} + 1}^{n}} + \color{#00f}{{1 \over \pars{2n - 1}2^{n}}} + {2n \over 2n - 1}\,\bracks{% I_{n + 1} - \int_{n}^{n + 1}{\dd x \over \pars{x^{2} + 1}^{n + 1}} } \end{align} The $\color{#00f}{\large blue}$ one is 'the leading term': $ {1 \over \pars{2n - 1}2^{n}} \sim {1 \over n2^{n + 1}} \sim \color{#00f}{\large{1 \over n\,2^{n}}} $