Find the value of int_(4)^(6)(5)/(8-x)dx. Write your answer as the logarithm of a single number in simplest form. Answer: ln(◻)

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Find the value of  int_(4)^(6)(5)/(8-x)dx. Write your answer as the logarithm of a single number in simplest form. Answer:  ln(◻)

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Perform Substitution: First, we perform the substitution u = 8 - x, which gives us du = -dx. This means that dx = -du. We also need to change the limits of integration. When x = 4, u = 8 - 4 = 4. When x = 6, u = 8 - 6 = 2.Rewrite Integral in terms of u: Now we can rewrite the integral in terms of u: ∫ from 4 to 6 of (5)/(8-x)dx = -∫ from 4 to 2 of (5/u)du. Notice that the limits of integration have switched because of the negative sign in the substitution. To correct this, we can take the negative sign outside the integral and reverse the limits back to their natural order.Integrate (5/u) with respect to u: The integral now becomes: -∫ from 4 to 2 of (5/u)du = ∫ from 2 to 4 of (5/u)du. Now we can integrate (5/u) with respect to u.Evaluate Integral from u = 2 to u = 4: The integral of (5/u)du is 5ln|u|. We evaluate this from u = 2 to u = 4: 5ln|u| from 2 to 4 = 5ln|4| - 5ln|2|.Simplify Expression Using Logarithm Properties: Now we simplify the expression using logarithm properties: 5ln|4| - 5ln|2| = 5(ln(4) - ln(2)) = 5ln(4/2) = 5ln(2).Final Answer: The final answer is 5ln(2), which is the logarithm of a single number in simplest form.

Find the value of $\int_{4}^{6} \frac{5}{8-x} d x$ . Write your answer as the logarithm of a single number in simplest form. $\newline$ Answer: $\ln (\square)$

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Find the value of $\int_{4}^{6} \frac{5}{8-x} d x$ . Write your answer as the logarithm of a single number in simplest form. $\newline$ Answer: $\ln (\square)$