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Question
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Solution
\[\text{We have}, \]
\[I = \int_{- \frac{\pi}{2}}^\frac{\pi}{2} x \cos^2 x\ d x\]
\[Let f\left( x \right) = x \cos^2 x\]
\[ \Rightarrow f\left( - x \right) = \left( - x \right) \cos^2 \left( - x \right)\]
\[ = - x \cos^2 x\]
\[ \therefore f\left( - x \right) = - f\left( x \right)\]
\[i . e . , f\left( x \right) \text{is odd function}\]
\[\text{We know that} \int_{- a}^a f\left( x \right) d x = 0 , \text{if }f\left( x \right) \text{is odd function} . \]
\[ \therefore I = \int_{- \frac{\pi}{2}}^\frac{\pi}{2} x \cos^2 x\ d x = 0\]
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