Question

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

Answer 1

$$\begin{gathered} \mathrm{Let}I={\int }_0^1\frac{1}{1+x^2}dx.\mathrm{Then}, \\ I={\left[{\tan }^{-1}x\right]}_0^1 \\ \Rightarrow I={\tan }^{-1}1-{\tan }^{-1}0 \\ \Rightarrow I=\frac{\mathrm{\pi }}{4}-0 \\ \Rightarrow I=\frac{\mathrm{\pi }}{4} \end{gathered}$$