Evaluate int_(0)^(4)(5e^(-0.5 x)-2x)dx and express the answer in simplest form. Answer:

Substitution Method Integration: We need to evaluate the integral of the function (5e^(-0.5x) - 2x) from 0 to 4. We will integrate each term separately. First, let's integrate 5e^(-0.5x). We can use the substitution method for this part of the integral. Let u = -0.5x, which implies du = -0.5dx or dx = -2du. The integral of e^u with respect to u is e^u. So, the integral of 5e^(-0.5x)dx is -10e^(-0.5x).Integrating -2x: Next, we integrate -2x with respect to x. The integral of x with respect to x is (1/2)x^2. So, the integral of -2x dx is -x^2.Combining Integrals: Now we combine the two integrals to get the integral of the entire function: ∫(5e^(-0.5x) - 2x)dx = -10e^(-0.5x) - x^2 + C, where C is the constant of integration.Evaluating Antiderivative: We evaluate this antiderivative from 0 to 4: [-10e^(-0.5x) - x^2] evaluated from 0 to 4. This gives us: [-10e^(-0.5*4) - 4^2] - [-10e^(-0.5*0) - 0^2].Plugging in Values: Now we plug in the values and simplify: [-10e^(-2) - 16] - [-10e^(0) - 0]. We know that e^(0) = 1, so this simplifies to: [-10/e^2 - 16] - [-10 - 0].Final Answer: Simplifying further, we get: -10/e^2 - 16 + 10.Finally, we combine like terms to get the final answer: -10/e^2 - 6.

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Evaluate
int_(0)^(4)(5e^(-0.5 x)-2x)dx and express the answer in simplest form.
Answer:

Evaluate $\int_{0}^{4}\left(5 e^{-0.5 x}-2 x\right) d x$ and express the answer in simplest form. $\newline$ Answer:

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Calculus

Evaluate definite integrals using the power rule

Full solution

Q. Evaluate

\int_{0}^{4}\left(5 e^{-0.5 x}-2 x\right) d x

and express the answer in simplest form.

\newline

Answer:

Substitution Method Integration: We need to evaluate the integral of the function $5e^{-0.5x} - 2x$ from $0$ to $4$ . We will integrate each term separately. $\newline$ First, let's integrate $5e^{-0.5x}$ . We can use the substitution method for this part of the integral. $\newline$ Let $u = -0.5x$ , which implies $du = -0.5dx$ or $dx = -2du$ . $\newline$ The integral of $e^u$ with respect to $u$ is $e^u$ . $\newline$ So, the integral of $0$ $0$ is $0$ $1$ .
Integrating $-2x$ : Next, we integrate $-2x$ with respect to $x$ . The integral of $x$ with respect to $x$ is $(1/2)x^2$ . $\newline$ So, the integral of $-2x \, dx$ is $-x^2$ .
Combining Integrals: Now we combine the two integrals to get the integral of the entire function: $\int(5e^{(-0.5x)} - 2x)dx = -10e^{(-0.5x)} - x^2 + C$ , where $C$ is the constant of integration.
Evaluating Antiderivative: We evaluate this antiderivative from $0$ to $4$ : $[-10e^{(-0.5x)} - x^2$ ] evaluated from $0$ to $4$ . This gives us: $[-10e^{(-0.5\cdot4)} - 4^2$ ] - [\(-10e^{( $-0$ . $5$ \cdot $0$ )} - $0$ ^ $2$ \]).
Plugging in Values: Now we plug in the values and simplify: $\newline$ [-10e^{-2} - 16\] - \[-10e^{0} - 0\]).$\newlineWe know that \$e^{0} = 1$, so this simplifies to:$\newline$$[\(-10$/e^{$2$} - $16$\] - \[-10 - 0\]).
Final Answer: Simplifying further, we get:$\newline$$-10/e^2 - 16 + 10$.
Final Answer: Simplifying further, we get:$\newline$$-10/e^2 - 16 + 10$.Finally, we combine like terms to get the final answer:$\newline$$-10/e^2 - 6$.