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Question
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Solution
\[Let\ I = \int_{- 1}^1 \frac{1}{x^2 + 2x + 5} d x . Then, \]
\[ I = \int_{- 1}^1 \frac{1}{\left( x^2 + 2x + 1 \right) + 4} d x\]
\[ \Rightarrow I = \int_{- 1}^1 \frac{1}{\left( x + 1 \right)^2 + 2^2} d x\]
\[ \Rightarrow I = \frac{1}{2} \left[ \tan^{- 1} \frac{\left( x + 1 \right)}{2} \right]_{- 1}^1 \]
\[ \Rightarrow I = \frac{1}{2}\left( \tan^{- 1} 1 - \tan^{- 1} 0 \right)\]
\[ \Rightarrow I = \frac{1}{2}\left( \frac{\pi}{4} \right)\]
\[ \Rightarrow I = \frac{\pi}{8}\]
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