$\"\"$

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Integration of Odd Functions:

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Let $\"\"$ be integrable on the closed interval $\"\"$ .

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If $\"\"$ is an odd function, then $\"\"$ .

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Need to verify $\"\"$ .

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If $\"\"$ is an odd function, then $\"\"$ .

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The integral is $\"\"$ .

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Apply Additive Interval Property: $\"\"$ . $\"\"$

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Apply special definite integrals $\"\"$ .

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$\"\"$ equation $\"\"$

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Consider $\"\"$ .

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Let $\"\"$ .

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$\"\"$

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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When $\"\"$ , $\"\"$ .

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When $\"\"$ , $\"\"$ .

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Substitute $\"\"$ and $\"\"$ in $\"\"$ .

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Apply limits: $\"\"$ to $\"\"$ .

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$\"\"$

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Since $\"\"$ is an odd function.

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$\"\"$ $\"\"$

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ in equation $\"\"$ .

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$\"\"$

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$\"\"$ . $\"\"$

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If $\"\"$ is an odd function, then $\"\"$ .