$\"\"$

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The integral is $\"\"$ .

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(a) Substitution where $\"\"$ .

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

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$\"\"$

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$\"\"$

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$\"\"$

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Substitute $\"\"$ .

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$\"\"$

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$\"\"$ .

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$\"\"$

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(b) Substitution where $\"\"$ .

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

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$\"\"$

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$\"\"$

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$\"\"$

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Substitute $\"\"$ .

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$\"\"$

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$\"\"$ .

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$\"\"$

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(c) Solve the integral $\"\"$ using Integration by parts.

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The formula for integration by parts is $\"\"$ .

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Here $\"\"$ and $\"\"$ .

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Consider $\"\"$

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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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Consider $\"\"$ .

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Apply integral on each side.

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$\"\"$

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$\"\"$ .

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

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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(d) Susbtitution for $\"\"$ .

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The integral $\"\"$ .

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$\"\"$

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Since $\"\"$ .

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$\"\"$

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Apply the formula : $\"\"$ .

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$\"\"$

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$\"\"$

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Therefore, $\"\"$ .

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The answers all are same but in different forms.

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Using trigonometric identities prove that all are in the same form.

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$\"\"$

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(a) $\"\"$ .

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(b) $\"\"$ .

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(c) $\"\"$ .

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(d) $\"\"$ .

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The answers all are same but in different forms.

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Using trigonometric identities prove that all are in the same form.