Evaluate int_(2)^(3)(4x^(2)-15 x-3)/(x-4)dx. Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).

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Evaluate  int_(2)^(3)(4x^(2)-15 x-3)/(x-4)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 - 15x - 3) / (x - 4). Dividing 4x^2 by x gives us 4x. Multiplying 4x by (x - 4) gives us 4x^2 - 16x. Subtracting this from the original numerator, we get 4x^2 - 15x - 3 - (4x^2 - 16x) = x + 3. So, the quotient is 4x + 1 and the remainder is x + 3. The integral can now be rewritten as the integral of 4x + 1 plus the integral of the remainder over x - 4.Integrate Simplified Expression Term by Term: Now we will integrate the simplified expression term by term. The integral of 4x is 2x^2, and the integral of 1 is x. For the remainder, we have the integral of (x + 3) / (x - 4). We can split this into two separate integrals: the integral of x / (x - 4) and the integral of 3 / (x - 4).Evaluate Antiderivative from 2 to 3: The integral of x / (x - 4) can be simplified by recognizing that the derivative of the denominator (x - 4) is 1, which matches the numerator x after adjusting for the constant shift. Therefore, the integral of x / (x - 4) is ln|x - 4|. The integral of 3 / (x - 4) is 3 times the integral of 1 / (x - 4), which is 3ln|x - 4|.Combine Logarithmic Terms: Combining all the terms, the integral becomes 2x^2 + x + ln|x - 4| + 3ln|x - 4|. Since we want to condense the logarithms into a single logarithm, we use the property that ln(a) + ln(b) = ln(ab) to combine the logarithmic terms. This gives us 2x^2 + x + ln|(x - 4)^4|.Simplify Integral at Upper and Lower Limits: Now we need to evaluate the antiderivative from 2 to 3. We plug in the upper limit of 3 into the antiderivative to get 2(3)^2 + 3 + ln|(3 - 4)^4|, which simplifies to 18 + 3 + ln|(-1)^4| = 21 + ln|1|. Then we plug in the lower limit of 2 into the antiderivative to get 2(2)^2 + 2 + ln|(2 - 4)^4|, which simplifies to 8 + 2 + ln|(-2)^4| = 10 + ln|16|.Subtract Values at Upper and Lower Limits: Subtracting the value at the lower limit from the value at the upper limit, we get (21 + ln|1|) - (10 + ln|16|). Since ln|1| is 0, this simplifies to 21 - 10 - ln|16| = 11 - ln|16|.Final Answer: The final answer in simplest form is 11 - ln|16|.

Evaluate $\int_{2}^{3} \frac{4 x^{2}-15 x-3}{x-4} d x$ . Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).

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Evaluate $\int_{2}^{3} \frac{4 x^{2}-15 x-3}{x-4} d x$ . Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).