Evaluate the following definite integrals: _ -1 ^11 x^2+2x+5 dx - Vidyadip

Question

Evaluate the following definite integrals:
$\int_{-1}^\limits{1}\frac{1}{\text{x}^2+2\text{x}+5}\text{ dx}$

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Answer

Let $\text{I}=\int_{-1}^\limits{1}\frac{1}{\text{x}^2+2\text{x}+5}\text{ dx}$ Then,
$\text{I}=\int_{-1}^\limits{1}\frac{1}{\big(\text{x}^2+2\text{x}+1\big)+4}\text{ dx}$
$\Rightarrow\text{I}=\int_{-1}^\limits{1}\frac{1}{(\text{x}+1)^2+2^2}\text{ dx}$
$\Rightarrow\text{I}=\frac{1}{2}\Big[\tan^{-1}\frac{(\text{x}+1)}{2}\Big]^1_{-1}$
$\Rightarrow\text{I}=\frac{1}{2}\Big(\frac{\pi}{4}\Big)$
$\Rightarrow\text{I}=\frac{\pi}{8}$

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