Evaluate the following definite integrals: _ -1 ^11 1+x^2 dx - Vidyadip

Question

Evaluate the following definite integrals:
$\int_{-1}^\limits{1}\frac{1}{1+\text{x}^2} \text{ dx}$

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Answer

Let $\text{I}=\int_{-1}^\limits{1}\frac{1}{1+\text{x}^2} \text{ dx}$ Then,
$\text{I}=\big[\tan^{-1}\text{x}\big]^1_{-1}$
$\Rightarrow\text{I}=\tan^{-1}1-\tan^{-1}(-1)$
$\Rightarrow\text{I}=\frac{\pi}{4}-\Big(-\frac{\pi}{4}\Big)$
$\Rightarrow\text{I}=\frac{\pi}{2}$

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