Evaluate int_(0)^(5)(7e^(-0.2 x)-4x)dx and express the answer in simplest form. Answer:

Break down into two integrals: Break down the integral into two separate integrals. We have the integral of a sum of two functions, which can be separated into the sum of two integrals: ∫(7e^(-0.2x) - 4x)dx = ∫7e^(-0.2x)dx - ∫4xdxEvaluate first integral: Evaluate the first integral ∫7e^(-0.2x)dx. To integrate 7e^(-0.2x), we use the fact that the integral of e^(ax) is (1/a)e^(ax), where a is a constant. So, ∫7e^(-0.2x)dx = -7/0.2 * e^(-0.2x) = -35e^(-0.2x)Evaluate second integral: Evaluate the second integral ∫4xdx. The integral of 4x with respect to x is 4x^2/2, which simplifies to 2x^2. So, ∫4xdx = 2x^2Combine results of integrals: Combine the results of the two integrals. The combined indefinite integral is: -35e^(-0.2x) + 2x^2 + C, where C is the constant of integration.Evaluate definite integral: Evaluate the definite integral from 0 to 5. We need to calculate the value of the combined integral at the upper limit (x=5) and subtract the value at the lower limit (x=0). For x=5: -35e^(-0.2*5) + 2*5^2 = -35e^(-1) + 50 For x=0: -35e^(-0.2*0) + 2*0^2 = -35e^(0) + 0 = -35 Now, subtract the value at x=0 from the value at x=5: (-35e^(-1) + 50) - (-35) = -35e^(-1) + 50 + 35Simplify the result: Simplify the result. -35e^(-1) + 50 + 35 = -35/e + 85 This is the exact, simplified answer for the definite integral.

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Evaluate
int_(0)^(5)(7e^(-0.2 x)-4x)dx and express the answer in simplest form.
Answer:

Evaluate $\int_{0}^{5}\left(7 e^{-0.2 x}-4 x\right) d x$ and express the answer in simplest form. $\newline$ Answer:

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Calculus

Evaluate definite integrals using the chain rule

Full solution

Q. Evaluate

\int_{0}^{5}\left(7 e^{-0.2 x}-4 x\right) d x

and express the answer in simplest form.

\newline

Answer:

Break down into two integrals: Break down the integral into two separate integrals. $\newline$ We have the integral of a sum of two functions, which can be separated into the sum of two integrals: $\newline$ \int(\(7e^{ $-0$ . $2$ x} - $4$ x)\,dx = \int $7$ e^{ $-0$ . $2$ x}\,dx - \int $4$ x\,dx
Evaluate first integral: Evaluate the first integral $\int 7e^{-0.2x}\,dx$ . To integrate $7e^{-0.2x}$ , we use the fact that the integral of $e^{ax}$ is $(1/a)e^{ax}$ , where $a$ is a constant. So, $\int 7e^{-0.2x}\,dx = -7/0.2 \cdot e^{-0.2x} = -35e^{-0.2x}$
Evaluate second integral: Evaluate the second integral $\int 4x\,dx$ . The integral of $4x$ with respect to $x$ is $\frac{4x^2}{2}$ , which simplifies to $2x^2$ . So, $\int 4x\,dx = 2x^2$
Combine results of integrals: Combine the results of the two integrals. $\newline$ The combined indefinite integral is: $\newline$ $-35e^{(-0.2x)} + 2x^2 + C$ , where $C$ is the constant of integration.
Evaluate definite integral: Evaluate the definite integral from $0$ to $5$ . We need to calculate the value of the combined integral at the upper limit $(x=5)$ and subtract the value at the lower limit $(x=0)$ . For $x=5$ : $-35e^{(-0.2\cdot 5)} + 2\cdot 5^2 = -35e^{(-1)} + 50$ For $x=0$ : $-35e^{(-0.2\cdot 0)} + 2\cdot 0^2 = -35e^{(0)} + 0 = -35$ Now, subtract the value at $x=0$ from the value at $x=5$ : $5$ $0$
Simplify the result: Simplify the result. $\newline$ $-35e^{-1} + 50 + 35 = -\frac{35}{e} + 85$ $\newline$ This is the exact, simplified answer for the definite integral.