$\"\"$

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(a) Prove that $\"\"$ .

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Consider the integral $\"\"$ .

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Rewrite the integral as $\"\"$ .

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Let $\"\"$ and $\"\"$ .

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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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Apply integral on each side.

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$\"\"$ .

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Apply integration by parts formula: $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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(b)

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Find $\"\"$ .

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The integral is $\"\"$ .

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From part (a) : $\"\"$ .

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In this case $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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Apply double angle formula: $\"\"$ .

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$\"\"$

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$\"\"$ .

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$\"\"$ .

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$\"\"$

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(c)

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Find $\"\"$

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The integral is $\"\"$ .

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From part (a) : $\"\"$ .

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In this case $\"\"$ .

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$\"\"$

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$\"\"$

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From part (b) substitute $\"\"$ .

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$\"\"$

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$\"\"$ , where $\"\"$ is a constant.

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$\"\"$ .

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$\"\"$

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(a) $\"\"$ .

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(b) $\"\"$ .

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(c) $\"\"$ .