Evaluate int_(0)^(2)(2e^(0.5 x)-2x)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 (2e^(0.5x) - 2x) from 0 to 2. This integral can be split into two separate integrals: int_(0)^(2)(2e^(0.5x) - 2x)dx = int_(0)^(2)2e^(0.5x)dx - int_(0)^(2)2xdxEvaluate First Integral: Evaluate the first integral. The first integral is int_(0)^(2)2e^(0.5x)dx. To solve this, we can use the substitution method or recognize that the antiderivative of e^(ax) is (1/a)e^(ax). Here, a = 0.5, so the antiderivative is (1/0.5)e^(0.5x) = 2e^(0.5x). Therefore, the integral becomes: int_(0)^(2)2e^(0.5x)dx = [2e^(0.5x)] from 0 to 2 Evaluating this at the bounds gives us: = 2e^(0.5*2) - 2e^(0.5*0) = 2e^1 - 2e^0 = 2e - 2Evaluate Second Integral: Evaluate the second integral. The second integral is int_(0)^(2)2xdx. The antiderivative of x is (1/2)x^2, so the antiderivative of 2x is 2*(1/2)x^2 = x^2. Therefore, the integral becomes: int_(0)^(2)2xdx = [x^2] from 0 to 2 Evaluating this at the bounds gives us: = 2^2 - 0^2 = 4 - 0 = 4Combine Results: Combine the results from Step 2 and Step 3. Now we combine the results of the two integrals to get the final answer: (2e - 2) - 4 = 2e - 2 - 4 = 2e - 6

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Evaluate
int_(0)^(2)(2e^(0.5 x)-2x)dx and express the answer in simplest form.
Answer:

Evaluate $\int_{0}^{2}\left(2 e^{0.5 x}-2 x\right) d x$ 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}^{2}\left(2 e^{0.5 x}-2 x\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 $2e^{0.5x} - 2x$ from $0$ to $2$ . This integral can be split into two separate integrals: $\newline$ $\int_{0}^{2}(2e^{0.5x} - 2x)\,dx = \int_{0}^{2}2e^{0.5x}\,dx - \int_{0}^{2}2x\,dx$
Evaluate First Integral: Evaluate the first integral. $\newline$ The first integral is $\int_{0}^{2} 2e^{0.5x}dx$ . To solve this, we can use the substitution method or recognize that the antiderivative of $e^{ax}$ is $(1/a)e^{ax}$ . Here, $a = 0.5$ , so the antiderivative is $(1/0.5)e^{0.5x} = 2e^{0.5x}$ . Therefore, the integral becomes: $\newline$ $\int_{0}^{2}2e^{0.5x}dx = [2e^{0.5x}]$ from $0$ to $2$ $\newline$ Evaluating this at the bounds gives us: $\newline$ $= 2e^{0.5\times 2} - 2e^{0.5\times 0}$ $\newline$ $= 2e^1 - 2e^0$ $\newline$ $e^{ax}$ $0$
Evaluate Second Integral: Evaluate the second integral. $\newline$ The second integral is $\int_{0}^{2} 2x \, dx$ . The antiderivative of $x$ is $(1/2)x^2$ , so the antiderivative of $2x$ is $2\times(1/2)x^2 = x^2$ . Therefore, the integral becomes: $\newline$ $\int_{0}^{2}2x\,dx = [x^2]$ from $0$ to $2$ $\newline$ Evaluating this at the bounds gives us: $\newline$ = $2^2 - 0^2$ $\newline$ = $4 - 0$ $\newline$ = $x$ $0$
Combine Results: Combine the results from Step $2$ and Step $3$ . $\newline$ Now we combine the results of the two integrals to get the final answer: $\newline$ $(2e - 2) - 4$ $\newline$ $= 2e - 2 - 4$ $\newline$ $= 2e - 6$