Find the value of int_(4)^(8)(10 dx)/(7-2x). Express your answer as a constant times ln 3. Answer: ◻ln 3

Identify the integral: Identify the integral to be solved. We need to evaluate the integral of the function (10 dx)/(7-2x) from 4 to 8.Simplify the integral: Simplify the integral. Let's use a substitution method to simplify the integral. We can let u = 7 - 2x, which means du = -2 dx or dx = -du/2.Change limits of integration: Change the limits of integration. When x = 4, u = 7 - 2(4) = -1. When x = 8, u = 7 - 2(8) = -9. So the new limits of integration are from u = -1 to u = -9.Rewrite integral in terms of u: Rewrite the integral in terms of u. The integral becomes: ∫ from -1 to -9 of (10 * -1/2 du)/u = -5 ∫ from -1 to -9 of (1/u) du.Evaluate the integral: Evaluate the integral. The integral of 1/u du is ln|u|. So we have: -5 [ln|u|] from -1 to -9.Apply limits of integration: Apply the limits of integration. -5 [ln|-9| - ln|-1|] = -5 [ln(9) - ln(1)] = -5 ln(9). Since ln(1) = 0, it simplifies to -5 ln(9).Simplify the expression: Simplify the expression. We know that ln(9) is the same as 2 ln(3), so the integral simplifies to: -5 * 2 ln(3) = -10 ln(3).Express answer as constant times ln(3): Express the answer as a constant times ln(3). The final answer is -10 ln(3).

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Find the value of
int_(4)^(8)(10 dx)/(7-2x). Express your answer as a constant times
ln 3.
Answer:
◻ln 3

Find the value of $\int_{4}^{8} \frac{10 d x}{7-2 x}$ . Express your answer as a constant times $\ln 3$ . $\newline$ Answer: $\square \ln 3$

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Calculus

Evaluate definite integrals using the power rule

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Q. Find the value of

\int_{4}^{8} \frac{10 d x}{7-2 x}

. Express your answer as a constant times

\ln 3

\newline

Answer:

\square \ln 3

Identify the integral: Identify the integral to be solved. $\newline$ We need to evaluate the integral of the function $\frac{10 \, dx}{7-2x}$ from $4$ to $8$ .
Simplify the integral: Simplify the integral. $\newline$ Let's use a substitution method to simplify the integral. We can let $u = 7 - 2x$ , which means $du = -2 dx$ or $dx = -\frac{du}{2}$ .
Change limits of integration: Change the limits of integration. When $x = 4$ , $u = 7 - 2(4) = -1$ . When $x = 8$ , $u = 7 - 2(8) = -9$ . So the new limits of integration are from $u = -1$ to $u = -9$ .
Rewrite integral in terms of u: Rewrite the integral in terms of u. $\newline$ The integral becomes: $\newline$ $\int_{-1}^{-9} \frac{10 \cdot (-\frac{1}{2}) \, du}{u} = -5 \int_{-1}^{-9} \frac{1}{u} \, du$ .
Evaluate the integral: Evaluate the integral. $\newline$ The integral of $\frac{1}{u} \, du$ is $\ln|u|$ . So we have: $\newline$ $-5 [\ln|u|]$ from $-1$ to $-9$ .
Apply limits of integration: Apply the limits of integration. $\newline$ $-5 [\ln|-9| - \ln|-1|] = -5 [\ln(9) - \ln(1)] = -5 \ln(9)$ . $\newline$ Since $\ln(1) = 0$ , it simplifies to $-5 \ln(9)$ .
Simplify the expression: Simplify the expression. $\newline$ We know that $\ln(9)$ is the same as $2 \ln(3)$ , so the integral simplifies to: $\newline$ $-5 \times 2 \ln(3) = -10 \ln(3)$ .
Express answer as constant times $\ln(3)$ : Express the answer as a constant times $\ln(3)$ . The final answer is $-10 \ln(3)$ .