\(I=\int\limits^1_0\frac{x^{2\left(n-2\right)}}{\left(1+x^2\right)^n}.xdx\)
Đặt \(1+x^2=t\Rightarrow xdx=\frac{1}{2}dt\)
\(\Rightarrow I=\frac{1}{2}\int\limits^2_1\frac{\left(t-1\right)^{n-2}}{t^n}dt=\frac{1}{2}\int\limits^2_1\left(\frac{t-1}{t}\right)^{n-2}.\frac{1}{t^2}dt=\frac{1}{2}\int\limits^2_1\left(1-\frac{1}{t}\right)^{n-2}.\frac{1}{t^2}dt\)
Đặt \(1-\frac{1}{t}=u\Rightarrow\frac{1}{t^2}dt=du\)
\(\Rightarrow I=\frac{1}{2}\int\limits^{\frac{1}{2}}_0u^{n-2}du=\frac{1}{2\left(n-1\right)}u^{n-1}|^{\frac{1}{2}}_0=\frac{1}{\left(n-1\right)2^n}\)
