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Evaluate int_(2)^(3)(4x^(2)-11 x-22)/(x-4)dx. Write your answer in simplest form with all logs condensed into a single logarithm (if necessary). Submit Answer

Evaluate ∫234x2−11x−22x−4dx \int_{2}^{3} \frac{4 x^{2}-11 x-22}{x-4} d x . Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).\newlineSubmit Answer

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Q. Evaluate ∫234x2−11x−22x−4dx \int_{2}^{3} \frac{4 x^{2}-11 x-22}{x-4} d x . Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).\newlineSubmit Answer
  1. Perform Polynomial Long Division: First, we will perform polynomial long division to simplify the integrand (4x2−11x−22)/(x−4)(4x^2 - 11x - 22) / (x - 4).
  2. Simplify Integrands: The division gives us:\newline4x2−11x−22=(x−4)(4x+5)+24x^2 - 11x - 22 = (x - 4)(4x + 5) + 2\newlineSo, (4x2−11x−22)/(x−4)=4x+5+2x−4(4x^2 - 11x - 22) / (x - 4) = 4x + 5 + \frac{2}{x - 4}
  3. Write Integral as Sum: Now we can write the integral as the sum of two simpler integrals: ∫23(4x+5+2x−4) dx\int_{2}^{3}(4x + 5 + \frac{2}{x - 4}) \, dx
  4. Integrate Each Term: We can integrate each term separately:\newline∫(4x+5) dx=2x2+5x+C\int(4x + 5) \, dx = 2x^2 + 5x + C\newline∫(2x−4)dx=2ln⁡∣x−4∣+C\int\left(\frac{2}{x - 4}\right) dx = 2\ln|x - 4| + C
  5. Evaluate Antiderivatives: Now we evaluate the antiderivatives from 22 to 33: \newline(2(3)2+5(3))−(2(2)2+5(2))+2ln⁡∣3−4∣−2ln⁡∣2−4∣(2(3)^2 + 5(3)) - (2(2)^2 + 5(2)) + 2\ln|3 - 4| - 2\ln|2 - 4|
  6. Perform Calculations: Perform the calculations:\newline(2(9)+15)−(2(4)+10)+2ln⁡∣−1∣−2ln⁡∣−2∣(2(9) + 15) - (2(4) + 10) + 2\ln|-1| - 2\ln|-2|\newline(18+15)−(8+10)+2ln⁡(1)−2ln⁡(2)(18 + 15) - (8 + 10) + 2\ln(1) - 2\ln(2)\newline33−18+0−2ln⁡(2)33 - 18 + 0 - 2\ln(2)\newline15−2ln⁡(2)15 - 2\ln(2)
  7. Final Answer: The final answer is: 15−2ln⁡(2)15 - 2\ln(2)

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