Find the value of int_(4)^(6)(3dx)/(x-8). Express your answer as a constant times ln 2. Answer: ◻ln 2

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Find the value of  int_(4)^(6)(3dx)/(x-8). Express your answer as a constant times  ln 2. Answer:  ◻ln 2

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Recognize standard form: Recognize the integral as a standard form. The integral $\int \frac{3dx}{x-8}$ can be recognized as a standard form of the integral $\int \frac{1}{u}du$, where $u = x - 8$.Perform substitution: Perform a substitution. Let $u = x - 8$, then $du = dx$. The integral becomes $\int \frac{3du}{u}$.Integrate using logarithm: Integrate using the natural logarithm. The integral of $\frac{1}{u}$ with respect to $u$ is $\ln|u|$. Therefore, the integral becomes $3\ln|u| + C$, where $C$ is the constant of integration.Substitute back for x: Substitute back for $x$. Since $u = x - 8$, the integral becomes $3\ln|x - 8| + C$.Evaluate definite integral: Evaluate the definite integral from 4 to 6. We need to evaluate $3\ln|x - 8|$ from $x = 4$ to $x = 6$. This gives us $3\ln|6 - 8| - 3\ln|4 - 8|$.Simplify expression: Simplify the expression. Simplify the expression to get $3\ln|-2| - 3\ln|-4|$. Since the natural logarithm of a positive number is the same as the natural logarithm of its absolute value, this simplifies to $3\ln(2) - 3\ln(4)$.Combine logarithm terms: Use logarithm properties to combine terms. Using the property of logarithms that $\ln(a) - \ln(b) = \ln(\frac{a}{b})$, we get $3\ln\left(\frac{2}{4}\right)$.Simplify fraction inside logarithm: Simplify the fraction inside the logarithm. The fraction $\frac{2}{4}$ simplifies to $\frac{1}{2}$, so the expression becomes $3\ln\left(\frac{1}{2}\right)$.Use logarithm property: Use the property of logarithms $\ln(a^{-1}) = -\ln(a)$. The expression $3\ln\left(\frac{1}{2}\right)$ is equivalent to $3(-\ln(2))$, which simplifies to $-3\ln(2)$.Express as constant times ln(2): Express the answer as a constant times $\ln(2)$. The final answer is $-3\ln(2)$, which is the value of the definite integral from 4 to 6 of $\frac{3dx}{x-8}$.

Find the value of $\int_{4}^{6} \frac{3 d x}{x-8}$ . Express your answer as a constant times $\ln 2$ . $\newline$ Answer: $\square \ln 2$

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Find the value of ∫463dxx−8 \int_{4}^{6} \frac{3 d x}{x-8} ∫46​x−83dx​. Express your answer as a constant times ln⁡2 \ln 2 ln2.\newlineAnswer: □ln⁡2 \square \ln 2 □ln2

Find the value of $\int_{4}^{6} \frac{3 d x}{x-8}$ . Express your answer as a constant times $\ln 2$ . $\newline$ Answer: $\square \ln 2$