$\"\"$

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The function is $\"\"$

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Let $\"\"$

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$\"\"$ (Apply the derivative)

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$\"\"$ (The derivative power rule)

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$\"\"$ (Divide each side by 2)

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$\"\"$ (Substitute $\"\"$ and $\"\"$ )

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$\"\"$

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$\"\"$ (The integral exponential rule)

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$\"\"$ (Substitute $\"\"$ )

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$\"\"$

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The integral is $\"\"$ .