Evaluate the definite integral _ 0 ^ 1 d x 1+x^ 2 - Vidyadip

Question

Evaluate the definite integral $\int_{0}^{1} \frac{d x}{1+x^{2}}$

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Answer

Let $I=\int_{0}^{1} \frac{d x}{1+x^{2}}$
$I=\int_{0}^{1} \frac{d x}{1+x^{2}}$
We know that
$\int \frac{d x}{a^{2}+x^{2}}=\frac{1}{a} \tan ^{-1} \frac{x}{a}+c $
Therefore,
$I = \left[\tan ^{-1} x\right]_{0}^{1} $
$= \tan^{-1}(1) - \tan^{-1}(0) = \frac{\pi}{4} - 0$
$= \frac{\pi}{4} $
$\therefore \int_{0}^{1} \frac{d x}{1+x^{2}}=\frac{\pi}{ 4}$

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