$\"\"$

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(a)

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The integral function is $\"\"$ .

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Evaluate the integral using integration by parts.

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Integration by parts formula : $\"\"$ .

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Assume $\"\"$ and $\"\"$ .

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Consider $\"\"$ .

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Differentiate on each side.

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$\"\"$ .

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$\"\"$ .

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Integrate on each side.

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$\"\"$

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$\"\"$ .

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Substitute the corresponding values in the formula.

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$\"\"$ .

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$\"\"$

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(b)

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The integral function is $\"\"$ .

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Assume $\"\"$ and $\"\"$ .

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Consider $\"\"$ .

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Differentiate on each side.

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$\"\"$ .

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$\"\"$ .

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Integrate on each side.

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$\"\"$

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$\"\"$ .

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Substitute the corresponding values in the formula.

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$\"\"$

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If $\"\"$ then $\"\"$ .

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$\"\"$

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Limits of the function $\"\"$ and $\"\"$ .

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$\"\"$

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Since $\"\"$ and $\"\"$ are inverse function $\"\"$ .

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$\"\"$ .

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$\"\"$

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(c)

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The integral is $\"\"$ .

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Geometric representation of the integral :

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$\"\"$

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In the above diagram :

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Blue color region represents $\"\"$ .

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Yellow color region repsents $\"\"$ .

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Rose color and blue colors combined region represents the region $\"\"$ .

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Rose color region represents $\"\"$ .

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$\"\"$

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(d)

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The integral function is $\"\"$ .

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The integral formula : $\"\"$ .

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If $\"\"$ then $\"\"$ .

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If $\"\"$ then $\"\"$ .

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If $\"\"$ then $\"\"$ .

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$\"\"$

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

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(a) $\"\"$ .

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(b) $\"\"$ .

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(c) Geometric representation of the integral is

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$\"\"$

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(d) $\"\"$ .