Evaluate int_(1)^(4)(4x^(2)-27 x-5)/(x-7)dx. Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).

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Evaluate  int_(1)^(4)(4x^(2)-27 x-5)/(x-7)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: Perform polynomial long division to simplify the integrand. We need to divide the polynomial 4x^2 - 27x - 5 by x - 7.Polynomial Long Division Calculation: Polynomial long division calculation. Dividing 4x^2 by x gives 4x. Multiplying (x - 7) by 4x gives 4x^2 - 28x. Subtracting this from 4x^2 - 27x gives x - 5. Dividing x by x gives 1. Multiplying (x - 7) by 1 gives x - 7. Subtracting this from x - 5 gives 2. So the division gives us 4x + 1 with a remainder of 2.Rewrite Integral with Result: Rewrite the integral with the result of the division. The integral becomes: ∫(4x + 1)dx + ∫(2/(x - 7))dx from 1 to 4.Integrate First Part: Integrate the first part of the integral. The integral of 4x + 1 with respect to x is 2x^2 + x.Integrate Second Part: Integrate the second part of the integral. The integral of 2/(x - 7) with respect to x is 2ln|x - 7|.Combine and Evaluate Definite Integral: Combine the results and evaluate the definite integral from 1 to 4. The integral from 1 to 4 is: (2x^2 + x) + 2ln|x - 7| evaluated from 1 to 4.Evaluate Antiderivative at Upper Limit: Evaluate the antiderivative at the upper limit x = 4. Plugging in x = 4 gives us: (2(4)^2 + 4) + 2ln|4 - 7| = (2(16) + 4) + 2ln|-3| = (32 + 4) + 2ln(3) = 36 + 2ln(3).Evaluate Antiderivative at Lower Limit: Evaluate the antiderivative at the lower limit x = 1. Plugging in x = 1 gives us: (2(1)^2 + 1) + 2ln|1 - 7| = (2(1) + 1) + 2ln|-6| = (2 + 1) + 2ln(6) = 3 + 2ln(6).Subtract Lower Limit from Upper Limit: Subtract the value at the lower limit from the value at the upper limit. (36 + 2ln(3)) - (3 + 2ln(6)) = 33 + 2ln(3) - 2ln(6).Simplify Using Properties of Logarithms: Simplify the expression using properties of logarithms. 2ln(3) - 2ln(6) can be simplified using the quotient rule of logarithms: ln(a) - ln(b) = ln(a/b). So, 2ln(3) - 2ln(6) = 2ln(3/6) = 2ln(1/2).Combine Logarithms into Single Logarithm: Combine the logarithms into a single logarithm. The final answer is 33 + 2ln(1/2).

Evaluate $\int_{1}^{4} \frac{4 x^{2}-27 x-5}{x-7} d x$ . Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).

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Evaluate $\int_{1}^{4} \frac{4 x^{2}-27 x-5}{x-7} d x$ . Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).