Evaluate int_(3)^(4)(4x^(2)-9x+1)/(x-2)dx. Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).

Question

Evaluate  int_(3)^(4)(4x^(2)-9x+1)/(x-2)dx. Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).

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Perform Polynomial Long Division: We will first perform polynomial long division to simplify the integrand (4x^2 - 9x + 1) / (x - 2).Simplify Integrand: Performing the division, we get:  (4x^2 - 9x + 1) : (x - 2) = 4x + 1 with a remainder of -1.  So, (4x^2 - 9x + 1) / (x - 2) = 4x + 1 - 1/(x - 2).Write Integral as Sum: Now we can write the integral as the sum of two simpler integrals:  ∫(4x + 1 - 1/(x - 2)) dx from x = 3 to x = 4.Integrate Each Term: We will integrate each term separately:  ∫(4x + 1) dx = 2x^2 + x + C  and  ∫(-1/(x - 2)) dx = -ln|x - 2| + C.Evaluate Definite Integral: Now we evaluate the definite integral from x = 3 to x = 4:  ∫(4x + 1 - 1/(x - 2)) dx from x = 3 to x = 4 = [2x^2 + x - ln|x - 2|] from x = 3 to x = 4.Upper Limit Calculation: Plugging in the upper limit x = 4:  2(4)^2 + 4 - ln|4 - 2| = 2(16) + 4 - ln|2| = 32 + 4 - ln(2).Lower Limit Calculation: Plugging in the lower limit x = 3:  2(3)^2 + 3 - ln|3 - 2| = 2(9) + 3 - ln|1| = 18 + 3 - ln(1) = 18 + 3 - 0 = 21.Subtract Values: Subtract the value at the lower limit from the value at the upper limit:  (32 + 4 - ln(2)) - (21) = 36 - ln(2) - 21 = 15 - ln(2).Final Answer: The final answer in simplest form with all logs condensed into a single logarithm is:  15 - ln(2).

Evaluate $\int_{3}^{4} \frac{4 x^{2}-9 x+1}{x-2} d x$ . Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).

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Evaluate ∫344x2−9x+1x−2dx \int_{3}^{4} \frac{4 x^{2}-9 x+1}{x-2} d x ∫34​x−24x2−9x+1​dx. Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).

Evaluate $\int_{3}^{4} \frac{4 x^{2}-9 x+1}{x-2} d x$ . Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).