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Q.
- Substitution: Let's do a substitution. Let , then , or .
- Change Limits: Change the limits of integration. When , . When , .
- Integral Transformation: Now substitute and change the integral: .
- Split Integral: Split the integral:
- Find Antiderivatives: Find the antiderivatives: from to .
- Plug in Limits: Plug in the limits of integration: \frac{\(1\)}{\(2\)} \times \left[\left(\frac{\(2\)}{\(3\)} \times \(9\)^{\frac{\(3\)}{\(2\)}} - \(2\) \times \(9\)^{\frac{\(1\)}{\(2\)}}\right) - \left(\frac{\(2\)}{\(3\)} \times \(1\)^{\frac{\(3\)}{\(2\)}} - \(2\) \times \(1\)^{\frac{\(1\)}{\(2\)}}\right)\right].
- Simplify Expression: Simplify the expression: \frac{}{} \times \left[\left(\frac{}{} \times - \times \right) - \left(\frac{}{} \times - \times \right)\right].
- Calculate Result: Do the math: $\frac{\(1\)}{\(2\)} \times \left[(\(18\) - \(6\)) - \left(\frac{\(2\)}{\(3\)} - \(2\)\right)\right] = \frac{\(1\)}{\(2\)} \times [\(12\) - \left(-\frac{\(4\)}{\(3\)}\right)] = \frac{\(1\)}{\(2\)} \times \left(\(12\) + \frac{\(4\)}{\(3\)}\right).
- Convert to Common Denominator: Convert to a common denominator and add: \(\frac{1}{2} \times \left(\frac{36}{3} + \frac{4}{3}\right) = \frac{1}{2} \times \frac{40}{3}.\)
- Final Calculation: Multiply by \(\frac{1}{2}\): \(\left(\frac{1}{2}\right) \times \left(\frac{40}{3}\right) = \frac{20}{3}\).
