Evaluate int_(0)^(2)(10e^(-0.5 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 two terms: 10e^(-0.5x) and 2x. We will integrate each term separately.Integrate First Term: Integrate the first term 10e^(-0.5x). The integral of e^(ax) with respect to x is (1/a)e^(ax), so the integral of 10e^(-0.5x) is -20e^(-0.5x). Calculation: ∫10e^(-0.5x)dx = -20e^(-0.5x) + CIntegrate Second Term: Integrate the second term 2x. The integral of x with respect to x is (1/2)x^2, so the integral of 2x is x^2. Calculation: ∫2xdx = x^2 + CCombine Integrals: Combine the results of the two integrals. The combined indefinite integral is -20e^(-0.5x) + x^2 + C.Evaluate Definite Integral: Evaluate the definite integral from 0 to 2. We need to calculate (-20e^(-0.5x) + x^2) evaluated at x=2 and subtract the value of the function evaluated at x=0. Calculation: [(-20e^(-0.5*2) + 2^2) - (-20e^(-0.5*0) + 0^2)]Perform Evaluation: Perform the evaluation and simplification. Calculation: [(-20e^(-1) + 4) - (-20e^(0) + 0)] Calculation: [(-20/e + 4) - (-20*1 + 0)] Calculation: [-20/e + 4 + 20] Calculation: 24 - 20/eWrite Final Answer: Write the final answer in simplest form. The final answer is 24 - 20/e.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Evaluate
int_(0)^(2)(10e^(-0.5 x)+2x)dx and express the answer in simplest form.
Answer:

Evaluate $\int_{0}^{2}\left(10 e^{-0.5 x}+2 x\right) d x$ and express the answer in simplest form. $\newline$ Answer:

View step-by-step help

Home

Math Problems

Calculus

Evaluate definite integrals using the chain rule

Full solution

Q. Evaluate

\int_{0}^{2}\left(10 e^{-0.5 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 two terms: $10e^{(-0.5x)}$ and $2x$ . We will integrate each term separately.
Integrate First Term: Integrate the first term $10e^{-0.5x}$ . The integral of $e^{ax}$ with respect to $x$ is $(1/a)e^{ax}$ , so the integral of $10e^{-0.5x}$ is $-20e^{-0.5x}$ . Calculation: $\int 10e^{-0.5x}dx = -20e^{-0.5x} + C$
Integrate Second Term: Integrate the second term $2x$ . The integral of $x$ with respect to $x$ is $(1/2)x^2$ , so the integral of $2x$ is $x^2$ . Calculation: $\int 2x\,dx = x^2 + C$
Combine Integrals: Combine the results of the two integrals. $\newline$ The combined indefinite integral is $-20e^{(-0.5x)} + x^2 + C$ .
Evaluate Definite Integral: Evaluate the definite integral from $0$ to $2$ . We need to calculate $(-20e^{-0.5x} + x^2)$ evaluated at $x=2$ and subtract the value of the function evaluated at $x=0$ . Calculation: $[(-20e^{-0.5\cdot 2} + 2^2) - (-20e^{-0.5\cdot 0} + 0^2)]$
Perform Evaluation: Perform the evaluation and simplification. $\newline$ Calculation: [(-20e^{-1} + 4) - (-20e^{0} + 0)]$\newlineCalculation: \$[(-20/e + 4) - (-20\cdot 1 + 0)]\(\newline$Calculation: \$[-20/e + 4 + 20]$\newline$Calculation: \$24 - 20/e\)
Write Final Answer: Write the final answer in simplest form.$\newline$The final answer is $24 - \frac{20}{e}$.