8.] Suppose int_(2)^(4)f(2x)dx=10. Then int_(4)^(8)f(u)du=

Given information: We are given that int_(2)^(4)f(2x)dx=10. To find int_(4)^(8)f(u)du, we need to perform a substitution to match the limits of integration. Let u = 2x, which implies that du = 2dx or dx = du/2.Perform substitution: Now we need to adjust the limits of integration. When x = 2, u = 2*2 = 4. When x = 4, u = 2*4 = 8. So the new limits of integration for u are from 4 to 8.Adjust limits: Substitute dx with du/2 in the integral and adjust the limits accordingly. int_(2)^(4)f(2x)dx = int_(4)^(8)f(u) * (1/2)duSubstitute and transform: Since we are given that int_(2)^(4)f(2x)dx=10, we can equate this to the transformed integral with the new variable and limits. 10 = (1/2) * int_(4)^(8)f(u)duEquating transformed integral: To find int_(4)^(8)f(u)du, we multiply both sides of the equation by 2. 2 * 10 = int_(4)^(8)f(u)duMultiply by 2: Perform the multiplication to solve for the integral. 20 = int_(4)^(8)f(u)du

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Suppose $\int_{2}^{4} f(2x) \, dx = 10$ . Then $\int_{4}^{8} f(u) \, du =$

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Calculus

Evaluate definite integrals using the power rule

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Q. Suppose

\int_{2}^{4} f(2x) \, dx = 10

. Then

\int_{4}^{8} f(u) \, du =

Given information: We are given that $\int_{2}^{4}f(2x)\,dx=10$ . To find $\int_{4}^{8}f(u)\,du$ , we need to perform a substitution to match the limits of integration. $\newline$ Let $u = 2x$ , which implies that $du = 2dx$ or $dx = \frac{du}{2}$ .
Perform substitution: Now we need to adjust the limits of integration. When $x = 2$ , $u = 2 \times 2 = 4$ . When $x = 4$ , $u = 2 \times 4 = 8$ . So the new limits of integration for $u$ are from $4$ to $8$ .
Adjust limits: Substitute $dx$ with $\frac{du}{2}$ in the integral and adjust the limits accordingly. $\int_{2}^{4}f(2x)dx = \int_{4}^{8}f(u) \cdot \left(\frac{1}{2}\right)du$
Substitute and transform: Since we are given that $\int_{2}^{4}f(2x)\,dx=10$ , we can equate this to the transformed integral with the new variable and limits. $\newline$ $10 = \left(\frac{1}{2}\right) \cdot \int_{4}^{8}f(u)\,du$
Equating transformed integral: To find $\int_{4}^{8}f(u)\,du$ , we multiply both sides of the equation by $2$ . $2 \times 10 = \int_{4}^{8}f(u)\,du$
Multiply by $2$ : Perform the multiplication to solve for the integral. $20 = \int_{4}^{8}f(u)\,du$