Evaluate int_(0)^(20)(9e^(0.2 x)+2)dx and express the answer in simplest form. Answer:

Identify Integral: Identify the integral to be solved. We need to evaluate the integral of the function 9e^(0.2x) + 2 with respect to x from 0 to 20.Break into Two: Break the integral into two separate integrals. The integral of a sum is the sum of the integrals, so we can write: ∫(9e^(0.2x) + 2)dx = ∫9e^(0.2x)dx + ∫2dxEvaluate First Integral: Evaluate the first integral ∫9e^(0.2x)dx. To integrate 9e^(0.2x), we can use the fact that the integral of e^(ax) is (1/a)e^(ax), where a is a constant. So, ∫9e^(0.2x)dx = (9/0.2)e^(0.2x) = 45e^(0.2x)Evaluate Second Integral: Evaluate the second integral ∫2dx. The integral of a constant is just the constant times the variable of integration, so: ∫2dx = 2xCombine Results: Combine the results of the two integrals. The combined indefinite integral is: 45e^(0.2x) + 2x + C, where C is the constant of integration.Evaluate Definite Integral: Evaluate the definite integral from 0 to 20. We need to calculate (45e^(0.2x) + 2x) evaluated at x=20 and subtract the value of the expression evaluated at x=0. F(20) = 45e^(0.2*20) + 2*20 F(0) = 45e^(0.2*0) + 2*0Perform Calculations: Perform the calculations for F(20) and F(0). F(20) = 45e^(4) + 40 F(0) = 45e^(0) + 0 = 45*1 + 0 = 45Subtract F(0): Subtract F(0) from F(20) to get the value of the definite integral. ∫(0 to 20)(9e^(0.2x) + 2)dx = F(20) - F(0) = (45e^(4) + 40) - 45Simplify Final Answer: Simplify the expression to get the final answer. ∫(0 to 20)(9e^(0.2x) + 2)dx = 45e^(4) + 40 - 45 = 45e^(4) - 5

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Evaluate
int_(0)^(20)(9e^(0.2 x)+2)dx and express the answer in simplest form.
Answer:

Evaluate $\int_{0}^{20}\left(9 e^{0.2 x}+2\right) d x$ and express the answer in simplest form. $\newline$ Answer:

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Calculus

Evaluate definite integrals using the chain rule

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Q. Evaluate

\int_{0}^{20}\left(9 e^{0.2 x}+2\right) d x

and express the answer in simplest form.

\newline

Answer:

Identify Integral: Identify the integral to be solved. $\newline$ We need to evaluate the integral of the function $9e^{0.2x} + 2$ with respect to $x$ from $0$ to $20$ .
Break into Two: Break the integral into two separate integrals. $\newline$ The integral of a sum is the sum of the integrals, so we can write: $\newline$ \int(\(9e^{ $0$ . $2$ x} + $2$ )\,dx = \int $9$ e^{ $0$ . $2$ x}\,dx + \int $2$ \,dx
Evaluate First Integral: Evaluate the first integral $\int 9e^{(0.2x)}\,dx$ . To integrate $9e^{(0.2x)}$ , we can use the fact that the integral of $e^{(ax)}$ is $(1/a)e^{(ax)}$ , where $a$ is a constant. So, $\int 9e^{(0.2x)}\,dx = (9/0.2)e^{(0.2x)} = 45e^{(0.2x)}$
Evaluate Second Integral: Evaluate the second integral $\int 2\,dx$ . The integral of a constant is just the constant times the variable of integration, so: $\int 2\,dx = 2x$
Combine Results: Combine the results of the two integrals. $\newline$ The combined indefinite integral is: $\newline$ $45e^{0.2x} + 2x + C$ , where $C$ is the constant of integration.
Evaluate Definite Integral: Evaluate the definite integral from $0$ to $20$ . We need to calculate $(45e^{0.2x} + 2x)$ evaluated at $x=20$ and subtract the value of the expression evaluated at $x=0$ . $F(20) = 45e^{0.2\times 20} + 2\times 20$ $F(0) = 45e^{0.2\times 0} + 2\times 0$
Perform Calculations: Perform the calculations for $F(20)$ and $F(0)$ . $\newline$ $F(20) = 45e^{4} + 40$ $\newline$ $F(0) = 45e^{0} + 0 = 45\times 1 + 0 = 45$
Subtract $F(0)$ : Subtract $F(0)$ from $F(20)$ to get the value of the definite integral. $\newline$ $\int_{0}^{20}(9e^{0.2x} + 2)dx = F(20) - F(0)$ $\newline$ $= (45e^{4} + 40) - 45$
Simplify Final Answer: Simplify the expression to get the final answer. $\newline$ $\int_{0}^{20}(9e^{0.2x} + 2)dx = 45e^{4} + 40 - 45$ $\newline$ $= 45e^{4} - 5$