Question:
$\int_{0}^{1} \frac{d x}{1+x^{2}}$
Solution:
Let $I=\int_{0}^{1} \frac{d x}{1+x^{2}}$
$\int \frac{d x}{1+x^{2}}=\tan ^{-1} x=\mathrm{F}(x)$
By second fundamental theorem of calculus, we obtain
$\begin{aligned} I &=\mathrm{F}(1)-\mathrm{F}(0) \\ &=\tan ^{-1}(1)-\tan ^{-1}(0) \\ &=\frac{\pi}{4} \end{aligned}$