Evaluate int_(6)^(e^(2)+5)(4x-21)/(x-5)dx. Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).

Question

Evaluate  int_(6)^(e^(2)+5)(4x-21)/(x-5)dx. Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).

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Simplify the integrand: Simplify the integrand by dividing the polynomial. We have the integral of a rational function, which can be simplified by polynomial division. The integrand (4x-21)/(x-5) can be divided to get a simpler expression.  (4x - 21) / (x - 5) = 4 - (1/(x - 5))  Now, we can write the integral as the sum of two simpler integrals:  ∫(4x - 21)/(x - 5) dx = ∫4 dx - ∫(1/(x - 5)) dxIntegrate the simplified expression: Integrate the simplified expression. The integral of a constant is just the constant times the variable, and the integral of 1/(x - 5) is the natural logarithm of the absolute value of (x - 5).  ∫4 dx = 4x ∫(1/(x - 5)) dx = ln|x - 5|  So the integral becomes:  ∫(4x - 21)/(x - 5) dx = 4x - ln|x - 5| + CEvaluate definite integral: Evaluate the definite integral from x=6 to x=e^(2)+5. We need to evaluate the antiderivative at the upper and lower limits and subtract the lower evaluation from the upper evaluation.  At x = e^(2) + 5: 4x - ln|x - 5| = 4(e^(2) + 5) - ln|e^(2) + 5 - 5| = 4e^(2) + 20 - ln|e^(2)|  At x = 6: 4x - ln|x - 5| = 4(6) - ln|6 - 5| = 24 - ln|1|  Now subtract the lower evaluation from the upper evaluation:  (4e^(2) + 20 - ln|e^(2)|) - (24 - ln|1|)  Since ln|1| is 0, this simplifies to:  4e^(2) + 20 - ln|e^(2)| - 24Simplify and write answer: Simplify the result and write the answer in simplest form. We can simplify the expression further by recognizing that ln|e^(2)| is simply 2, since the natural logarithm is the inverse of the exponential function.  4e^(2) + 20 - 2 - 24 = 4e^(2) - 6  This is the value of the definite integral from x=6 to x=e^(2)+5.

Evaluate $\int_{6}^{e^{2}+5} \frac{4 x-21}{x-5} d x$ . Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).

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Evaluate ∫6e2+54x−21x−5dx \int_{6}^{e^{2}+5} \frac{4 x-21}{x-5} d x ∫6e2+5​x−54x−21​dx. Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).

Evaluate $\int_{6}^{e^{2}+5} \frac{4 x-21}{x-5} d x$ . Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).