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Evaluate
int_(0)^(8)(8e^(-0.25 x)-6)dx and express the answer in simplest form.
Answer:

Evaluate $\int_{0}^{8}\left(8 e^{-0.25 x}-6\right) d x$ and express the answer in simplest form. $\newline$ Answer:

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Evaluate rational expressions II

Full solution

Q. Evaluate

\int_{0}^{8}\left(8 e^{-0.25 x}-6\right) d x

and express the answer in simplest form.

\newline

Answer:

Identify Integral: Identify the integral to be evaluated. $\newline$ We need to evaluate the integral of the function $8e^{-0.25x} - 6$ with respect to $x$ from $0$ to $8$ .
Break Down Integral: Break down the integral into two separate integrals. $\newline$ The integral of a sum of functions is the sum of the integrals of each function. Therefore, we can write: $\newline$ $\int(8e^{-0.25x} - 6)\,dx = \int 8e^{-0.25x}\,dx - \int 6\,dx$
Evaluate First Integral: Evaluate the first integral $\int 8e^{(-0.25x)}dx$ . To integrate $8e^{(-0.25x)}$ , we use the substitution method. Let $u = -0.25x$ , then $du = -0.25dx$ , or $dx = -4du$ . The limits of integration also change with the substitution. When $x = 0$ , $u = 0$ , and when $x = 8$ , $u = -2$ . The integral becomes $-32\int e^u du$ from $u = 0$ to $u = -2$ .
Evaluate Second Integral: Evaluate the second integral $\int 6\,dx$ . The integral of a constant is the constant times the variable of integration. Therefore, $\int 6\,dx = 6x$ . We evaluate this from $0$ to $8$ , which gives us $6(8) - 6(0) = 48$ .
Evaluate $-32\int e^u du$ : Evaluate the integral $-32\int e^u du$ from $u = 0$ to $u = -2$ . The integral of $e^u$ with respect to $u$ is $e^u$ . Therefore, $-32\int e^u du = -32(e^u)$ . We evaluate this from $u = 0$ to $u = -2$ , which gives us $-32\int e^u du$ $0$ .
Combine Results: Combine the results from Step $4$ and Step $5$ . $\newline$ We have $-32(e^{-2} - 1) + 48$ . $\newline$ Now we need to calculate $e^{-2}$ and simplify the expression. $\newline$ $e^{-2}$ is approximately $0.1353$ . $\newline$ So, $-32(0.1353 - 1) + 48 = -32(-0.8647) + 48 = 27.6704 + 48 = 75.6704$ .
Express Answer: Express the answer in simplest form. $\newline$ The simplest form of the answer is a decimal rounded to an appropriate number of significant figures. Since we have already calculated the numerical value, the answer is $75.6704$ .

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