Evaluate int_(0)^(5)(3e^(0.2 x)-2x)dx and express the answer in simplest form. Answer:

Identify Terms: Identify the two separate terms in the integral. We have the integral of the sum of two functions: 3e^(0.2x) and -2x. We can integrate each term separately.Integrate 3e^(0.2x): Integrate the first term 3e^(0.2x). To integrate 3e^(0.2x), we use the fact that the integral of e^(ax) is (1/a)e^(ax), where a is a constant. So, the integral of 3e^(0.2x) is (3/0.2)e^(0.2x) = 15e^(0.2x).Integrate -2x: Integrate the second term -2x. The integral of -2x with respect to x is -x^2.Combine Integrals: Combine the integrals of the two terms. The integral of the function 3e^(0.2x) - 2x is 15e^(0.2x) - x^2.Evaluate Definite Integral: Evaluate the definite integral from 0 to 5. We need to calculate (15e^(0.2x) - x^2) evaluated at x=5 and subtract the value of the function evaluated at x=0. At x=5: 15e^(0.2*5) - 5^2 = 15e^1 - 25. At x=0: 15e^(0.2*0) - 0^2 = 15e^0 - 0 = 15 - 0 = 15.Subtract Values: Subtract the value at the lower limit from the value at the upper limit. (15e^1 - 25) - (15) = 15e - 25 - 15 = 15e - 40.

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Evaluate
int_(0)^(5)(3e^(0.2 x)-2x)dx and express the answer in simplest form.
Answer:

Evaluate $\int_{0}^{5}\left(3 e^{0.2 x}-2 x\right) d x$ and express the answer in simplest form. $\newline$ Answer:

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Calculus

Evaluate definite integrals using the chain rule

Full solution

Q. Evaluate

\int_{0}^{5}\left(3 e^{0.2 x}-2 x\right) d x

and express the answer in simplest form.

\newline

Answer:

Identify Terms: Identify the two separate terms in the integral. $\newline$ We have the integral of the sum of two functions: $3e^{0.2x}$ and $-2x$ . We can integrate each term separately.
Integrate $3e^{0.2x}$ : Integrate the first term $3e^{0.2x}$ . $\newline$ To integrate $3e^{0.2x}$ , we use the fact that the integral of $e^{ax}$ is $(1/a)e^{ax}$ , where $a$ is a constant. $\newline$ So, the integral of $3e^{0.2x}$ is $(3/0.2)e^{0.2x} = 15e^{0.2x}$ .
Integrate $-2x$ : Integrate the second term $-2x$ . The integral of $-2x$ with respect to $x$ is $-x^2$ .
Combine Integrals: Combine the integrals of the two terms. $\newline$ The integral of the function $3e^{0.2x} - 2x$ is $15e^{0.2x} - x^2$ .
Evaluate Definite Integral: Evaluate the definite integral from $0$ to $5$ . We need to calculate $(15e^{0.2x} - x^2)$ evaluated at $x=5$ and subtract the value of the function evaluated at $x=0$ . At $x=5$ : $15e^{0.2\times 5} - 5^2 = 15e^1 - 25$ . At $x=0$ : $15e^{0.2\times 0} - 0^2 = 15e^0 - 0 = 15 - 0 = 15$ .
Subtract Values: Subtract the value at the lower limit from the value at the upper limit. $\newline$ $(15e^1 - 25) - (15) = 15e - 25 - 15 = 15e - 40$ .