Evaluate int_(0)^(4)(9e^(0.25 x)+4x)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^(0.25x) + 4x with respect to x from 0 to 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: ∫(9e^(0.25x) + 4x)dx = ∫9e^(0.25x)dx + ∫4xdxEvaluate First Integral: Evaluate the first integral ∫9e^(0.25x)dx. To integrate 9e^(0.25x), we can use the substitution method: Let u = 0.25x, then du = 0.25dx, or dx = 4du. The integral becomes: ∫9e^(u) * 4du = 36∫e^(u)du The integral of e^(u) is e^(u), so we have: 36∫e^(u)du = 36e^(u) + C Substituting back for u, we get: 36e^(0.25x) + CEvaluate Second Integral: Evaluate the second integral ∫4xdx. The integral of 4x with respect to x is 2x^2, so we have: ∫4xdx = 2x^2 + CCombine Results: Combine the results of the two integrals. The combined indefinite integral is: 36e^(0.25x) + 2x^2 + CEvaluate Definite Integral: Evaluate the definite integral from 0 to 4. We need to calculate the value of the combined integral at the upper limit (x=4) and subtract the value at the lower limit (x=0). For x=4: 36e^(0.25*4) + 2*4^2 = 36e^(1) + 2*16 = 36e + 32 For x=0: 36e^(0.25*0) + 2*0^2 = 36e^(0) + 0 = 36 Now subtract the value at x=0 from the value at x=4: (36e + 32) - 36 = 36e + 32 - 36 = 36e - 4Write Final Answer: Write the final answer. The value of the definite integral from 0 to 4 is 36e - 4.

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Evaluate
int_(0)^(4)(9e^(0.25 x)+4x)dx and express the answer in simplest form.
Answer:

Evaluate $\int_{0}^{4}\left(9 e^{0.25 x}+4 x\right) d x$ 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}^{4}\left(9 e^{0.25 x}+4 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 $9e^{0.25x} + 4x$ with respect to $x$ from $0$ to $4$ .
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: $\newline$ $\int(9e^{0.25x} + 4x)dx = \int 9e^{0.25x}dx + \int 4xdx$
Evaluate First Integral: Evaluate the first integral $\int 9e^{(0.25x)}dx$ . To integrate $9e^{(0.25x)}$ , we can use the substitution method: Let $u = 0.25x$ , then $du = 0.25dx$ , or $dx = 4du$ . The integral becomes: $\int 9e^{(u)} \cdot 4du = 36\int e^{(u)}du$ The integral of $e^{(u)}$ is $e^{(u)}$ , so we have: $36\int e^{(u)}du = 36e^{(u)} + C$ Substituting back for $u$ , we get: $9e^{(0.25x)}$ $0$
Evaluate Second Integral: Evaluate the second integral $\int 4x \, dx$ . The integral of $4x$ with respect to $x$ is $2x^2$ , so we have: $\int 4x \, dx = 2x^2 + C$
Combine Results: Combine the results of the two integrals. $\newline$ The combined indefinite integral is: $\newline$ $36e^{0.25x} + 2x^2 + C$
Evaluate Definite Integral: Evaluate the definite integral from $0$ to $4$ . We need to calculate the value of the combined integral at the upper limit ( $x=4$ ) and subtract the value at the lower limit ( $x=0$ ). For $x=4$ : $36e^{(0.25\cdot4)} + 2\cdot4^2 = 36e^{1} + 2\cdot16 = 36e + 32$ For $x=0$ : $36e^{(0.25\cdot0)} + 2\cdot0^2 = 36e^{0} + 0 = 36$ Now subtract the value at $x=0$ from the value at $x=4$ : $4$ $0$
Write Final Answer: Write the final answer. $\newline$ The value of the definite integral from $0$ to $4$ is $36e - 4$ .