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

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

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Identify Integral: Identify the integral to be evaluated. 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. The integral of a sum of functions is the sum of the integrals of each function. Therefore, we can write: ∫(8e^(-0.25x) - 6)dx = ∫8e^(-0.25x)dx - ∫6dxEvaluate First Integral: Evaluate the first integral ∫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∫e^u du from u = 0 to u = -2.Evaluate Second Integral: Evaluate the second integral ∫6dx. The integral of a constant is the constant times the variable of integration. Therefore, ∫6dx = 6x. We evaluate this from 0 to 8, which gives us 6(8) - 6(0) = 48.Evaluate -32∫e^u du: Evaluate the integral -32∫e^u du from u = 0 to u = -2. The integral of e^u with respect to u is e^u. Therefore, -32∫e^u du = -32(e^u). We evaluate this from u = 0 to u = -2, which gives us -32(e^(-2) - e^(0)) = -32(e^(-2) - 1).Combine Results: Combine the results from Step 4 and Step 5. We have -32(e^(-2) - 1) + 48. Now we need to calculate e^(-2) and simplify the expression. e^(-2) is approximately 0.1353. So, -32(0.1353 - 1) + 48 = -32(-0.8647) + 48 = 27.6704 + 48 = 75.6704.Express Answer: Express the answer in simplest form. 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.

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