Question

$\underset{0}{\overset{1}{\int }}\frac{2x}{1+x^4}dx$

Answer 1

$$\begin{gathered} \mathrm{Let}I={\int }_0^1\frac{2x}{1+x^4}\mathit{d}x\mathit{.} \\ \mathrm{Let}{x}^2=t.\mathrm{Then},2xdx=dt \\ \mathrm{When}x=0,t=0\mathrm{and}x=1,t=1 \\ \therefore I={\int }_0^1\frac{2x}{1+x^4}\mathit{d}x \\ \Rightarrow I={\int }_0^1\frac{1}{1+t^2}\mathit{d}t \\ \Rightarrow I={\left[{\tan }^{-1}t\right]}_0^1 \\ \Rightarrow I={\tan }^{-1}1-{\tan }^{-1}0 \\ \Rightarrow I=\frac{\mathrm{\pi }}{4} \end{gathered}$$