Evaluate int_(0)^(4)(3e^(-0.25 x)+8)dx and express the answer in simplest form. Answer:

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Evaluate  int_(0)^(4)(3e^(-0.25 x)+8)dx and express the answer in simplest form. Answer:

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Identify Function: Identify the function to integrate. We need to integrate the function f(x) = 3e^(-0.25x) + 8 over the interval [0, 4].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: ∫(3e^(-0.25x) + 8)dx = ∫3e^(-0.25x)dx + ∫8dxIntegrate First Part: Integrate the first part ∫3e^(-0.25x)dx. To integrate 3e^(-0.25x), we use the substitution method. Let u = -0.25x, then du = -0.25dx, or dx = -4du. The integral becomes -12∫e^u du, which evaluates to -12e^u + C. Substituting back for u, we get -12e^(-0.25x) + C.Integrate Second Part: Integrate the second part ∫8dx. The integral of a constant is the constant times the variable of integration, so ∫8dx = 8x + C.Combine Integrals: Combine the two integrals. The combined integral is -12e^(-0.25x) + 8x + C.Evaluate Definite Integral: Evaluate the definite integral from 0 to 4. We need to calculate the combined integral at the upper limit x = 4 and subtract the value of the combined integral at the lower limit x = 0. F(4) = -12e^(-0.25*4) + 8*4 F(0) = -12e^(-0.25*0) + 8*0Calculate F Values: Calculate F(4) and F(0). F(4) = -12e^(-1) + 32 F(0) = -12e^(0) + 0 F(0) simplifies to -12 + 0, since e^0 = 1.Subtract F(0) from F(4): Subtract F(0) from F(4) to get the definite integral. ∫(3e^(-0.25x) + 8)dx from 0 to 4 = F(4) - F(0) = (-12e^(-1) + 32) - (-12) = -12e^(-1) + 32 + 12Simplify Final Answer: Simplify the expression to get the final answer. -12e^(-1) + 32 + 12 simplifies to -12/e + 44.

Evaluate $\int_{0}^{4}\left(3 e^{-0.25 x}+8\right) d x$ and express the answer in simplest form. $\newline$ Answer:

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Evaluate $\int_{0}^{4}\left(3 e^{-0.25 x}+8\right) d x$ and express the answer in simplest form. $\newline$ Answer: