Evaluate the first integral

Oct 13, 2025 by windailys.com 106 views

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Introduction

In this article, we will evaluate the definite integral $\int_0^2\left(5 e^{-0.5 x}+8\right) d x$ and express the answer in its simplest form. The definite integral is a fundamental concept in calculus, and it has numerous applications in various fields, including physics, engineering, and economics.

What is a Definite Integral?

A definite integral is a mathematical operation that calculates the area under a curve or the accumulation of a quantity over a given interval. It is denoted by the symbol $\int_a^b f(x) dx$ , where $f(x)$ is the function being integrated, and $a$ and $b$ are the limits of integration.

Evaluating the Definite Integral

To evaluate the definite integral $\int_0^2\left(5 e^{-0.5 x}+8\right) d x$ , we can use the following steps:

Break down the integral: The given integral can be broken down into two separate integrals: $\int_0^2 5 e^{-0.5 x} dx$ and $\int_0^2 8 dx$ .
Evaluate the first integral: To evaluate the first integral, we can use the formula for the integral of an exponential function: $\int e^{ax} dx = \frac{1}{a} e^{ax} + C$ . In this case, $a = -0.5$ , so we have:

\int_0^2 5 e^{-0.5 x} dx = 5 \left[ \frac{1}{-0.5} e^{-0.5 x} \right]_0^2

Simplifying the expression, we get:

\int_0^2 5 e^{-0.5 x} dx = -10 \left[ e^{-0.5 x} \right]_0^2

Evaluating the expression at the limits of integration, we get:

\int_0^2 5 e^{-0.5 x} dx = -10 \left[ e^{-1} - 1 \right]

Evaluate the second integral: The second integral is a simple integral of a constant function:

\int_0^2 8 dx = 8 \left[ x \right]_0^2

Evaluating the expression at the limits of integration, we get:

\int_0^2 8 dx = 16

Combining the Results

Now that we have evaluated both integrals, we can combine the results to get the final answer:

\int_0^2\left(5 e^{-0.5 x}+8\right) d x = -10 \left[ e^{-1} - 1 \right] + 16

Simplifying the expression, we get:

\int_0^2\left(5 e^{-0.5 x}+8\right) d x = -10 e^{-1} + 10 + 16

\int_0^2\left(5 e^{-0.5 x}+8\right) d x = -10 e^{-1} + 26

Conclusion

In this article, we evaluated the definite integral $\int_0^2\left(5 e^{-0.5 x}+8\right) d x$ and expressed the answer in its simplest form. The final answer is $-10 e^{-1} + 26$ . This result can be used in various applications, including physics, engineering, and economics.

Final Answer

The final answer is $\boxed{-10 e^{-1} + 26}$ .

References

[1] Calculus: Early Transcendentals, James Stewart, 8th edition.
[2] Calculus, Michael Spivak, 4th edition.

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Introduction

In our previous article, we evaluated the definite integral $\int_0^2\left(5 e^{-0.5 x}+8\right) d x$ and expressed the answer in its simplest form. In this article, we will answer some frequently asked questions related to evaluating definite integrals.

Q&A

Q: What is a definite integral?

A: A definite integral is a mathematical operation that calculates the area under a curve or the accumulation of a quantity over a given interval. It is denoted by the symbol $\int_a^b f(x) dx$ , where $f(x)$ is the function being integrated, and $a$ and $b$ are the limits of integration.

Q: How do I evaluate a definite integral?

A: To evaluate a definite integral, you can use the following steps:

Break down the integral: If the integral is a sum of two or more functions, break it down into separate integrals.
Evaluate each integral: Use the formulas for the integrals of basic functions, such as exponential, trigonometric, and polynomial functions.
Combine the results: Add or subtract the results of each integral to get the final answer.

Q: What are some common formulas for definite integrals?

A: Some common formulas for definite integrals include:

Exponential function: $\int e^{ax} dx = \frac{1}{a} e^{ax} + C$
Trigonometric function: $\int \sin(ax) dx = -\frac{1}{a} \cos(ax) + C$
Polynomial function: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$

Q: How do I handle limits of integration?

A: When evaluating a definite integral, you need to handle the limits of integration carefully. Make sure to evaluate the function at the upper and lower limits of integration and subtract the results.

Q: What are some common mistakes to avoid when evaluating definite integrals?

A: Some common mistakes to avoid when evaluating definite integrals include:

Forgetting to break down the integral: Make sure to break down the integral into separate integrals if it is a sum of two or more functions.
Using the wrong formula: Use the correct formula for the integral of the function being integrated.
Not handling limits of integration correctly: Make sure to evaluate the function at the upper and lower limits of integration and subtract the results.

Conclusion

Evaluating definite integrals is an important concept in calculus, and it has numerous applications in various fields. By following the steps outlined in this article and avoiding common mistakes, you can evaluate definite integrals with confidence.

Final Answer

The final answer is $\boxed{-10 e^{-1} + 26}$ .

References

[1] Calculus: Early Transcendentals, James Stewart, 8th edition.
[2] Calculus, Michael Spivak, 4th edition.