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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

$\begin{aligned} I &=\mathrm{F}(1)-\mathrm{F}(0) \\ &=\tan ^{-1}(1)-\tan ^{-1}(0) \\ &=\frac{\pi}{4} \end{aligned}$

 

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