Mathematics

Evaluate the following definite integral: 
$$\displaystyle \int_{-1}^{1}\dfrac {1}{x^2 +2x+5}dx$$


SOLUTION
$$I=\displaystyle \int_{-1}^1\dfrac {1}{x^2+2x+5}dx $$

$$=\displaystyle \int_{-1}^1\dfrac {1}{(x+1)^2 +2^2}dx$$

Using, $$\displaystyle \int \dfrac {dx}{\sqrt {x^2+a^2} }=\tan^{-1}\dfrac xa$$, we get

$$=\dfrac {1}{2}\left [\tan^{-1} \dfrac {x+1}{2}\right]_{-1}^1 $$

$$=\dfrac {1}{2}(\tan^{-1} 1-\tan^{-1}0)$$

$$=\dfrac {\pi}{8}$$

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Subjective Medium Published on 17th 09, 2020
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