Evaluate $\int_0^5\left(10 E^{-0.2 X}-3\right) D X$ And Express The Answer In Simplest Form.

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Introduction

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In this article, we will evaluate the definite integral of the function $f(x) = 10 e^{-0.2 x}-3$ from $x = 0$ to $x = 5$. The definite integral is a fundamental concept in calculus, and it has numerous applications in various fields, including physics, engineering, and economics. To evaluate the definite integral, we will use the fundamental theorem of calculus, which states that the definite integral of a function $f(x)$ from $a$ to $b$ is equal to the antiderivative of $f(x)$ evaluated at $b$ minus the antiderivative of $f(x)$ evaluated at $a$.

Step 1: Find the Antiderivative of $f(x)$

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To evaluate the definite integral, we need to find the antiderivative of $f(x) = 10 e^{-0.2 x}-3$. The antiderivative of $f(x)$ is denoted by $F(x)$ and is defined as:

$$F(x) = \int f(x) d x$$

To find the antiderivative of $f(x)$, we will use the following rules:

• The antiderivative of $e^{ax}$ is $\frac{1}{a} e^{ax}$
• The antiderivative of $c$ is $cx$, where $c$ is a constant

Using these rules, we can find the antiderivative of $f(x)$ as follows:

$$F(x) = \int (10 e^{-0.2 x}-3) d x$$

$$F(x) = \int 10 e^{-0.2 x} d x - \int 3 d x$$

$$F(x) = \frac{10}{-0.2} e^{-0.2 x} - 3x + C$$

$$F(x) = -50 e^{-0.2 x} - 3x + C$$

where $C$ is the constant of integration.

Step 2: Evaluate the Antiderivative at the Limits of Integration

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Now that we have found the antiderivative of $f(x)$, we can evaluate it at the limits of integration, $x = 0$ and $x = 5$. We will use the following formula:

$$\int_a^b f(x) d x = F(b) - F(a)$$

where $F(x)$ is the antiderivative of $f(x)$.

Evaluating the antiderivative at $x = 0$, we get:

$$F(0) = -50 e^{-0.2 (0)} - 3(0) + C$$

$$F(0) = -50 + C$$

Evaluating the antiderivative at $x = 5$, we get:

$$F(5) = -50 e^{-0.2 (5)} - 3(5) + C$$

$$F(5) = -50 e^{-1} - 15 + C$$

Step 3: Simplify the Expression

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Now that we have evaluated the antiderivative at the limits of integration, we can simplify the expression by combining the terms.

$$\int_0^5 (10 e^{-0.2 x}-3) d x = F(5) - F(0)$$

$$\int_0^5 (10 e^{-0.2 x}-3) d x = (-50 e^{-1} - 15 + C) - (-50 + C)$$

$$\int_0^5 (10 e^{-0.2 x}-3) d x = -50 e^{-1} - 15 + 50$$

$$\int_0^5 (10 e^{-0.2 x}-3) d x = -50 e^{-1} + 35$$

Conclusion

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In this article, we evaluated the definite integral of the function $f(x) = 10 e^{-0.2 x}-3$ from $x = 0$ to $x = 5$. We used the fundamental theorem of calculus to find the antiderivative of $f(x)$ and evaluated it at the limits of integration. Finally, we simplified the expression to obtain the final answer.

The final answer is $\boxed{-50 e^{-1} + 35}$.

Final Answer

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The final answer is $\boxed{-50 e^{-1} + 35}$.

References

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• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart

Tags

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• Calculus
• Definite Integral
• Antiderivative
• Fundamental Theorem of Calculus
• Exponential Function
• Trigonometric Function
  

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Introduction

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In our previous article, we evaluated the definite integral of the function $f(x) = 10 e^{-0.2 x}-3$ from $x = 0$ to $x = 5$. In this article, we will answer some frequently asked questions related to evaluating definite integrals.

Q: What is a definite integral?

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A: A definite integral is a mathematical concept that represents the area under a curve between two points. It is denoted by $\int_a^b f(x) d x$ and is used to find the area between a curve and the x-axis.

Q: How do I evaluate a definite integral?

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A: To evaluate a definite integral, you need to follow these steps:

1. Find the antiderivative of the function $f(x)$.
2. Evaluate the antiderivative at the upper limit of integration, $b$.
3. Evaluate the antiderivative at the lower limit of integration, $a$.
4. Subtract the value of the antiderivative at $a$ from the value of the antiderivative at $b$.

Q: What is the fundamental theorem of calculus?

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A: The fundamental theorem of calculus is a mathematical concept that states that the definite integral of a function $f(x)$ from $a$ to $b$ is equal to the antiderivative of $f(x)$ evaluated at $b$ minus the antiderivative of $f(x)$ evaluated at $a$. It is denoted by:

$$\int_a^b f(x) d x = F(b) - F(a)$$

where $F(x)$ is the antiderivative of $f(x)$.

Q: How do I find the antiderivative of a function?

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A: To find the antiderivative of a function, you need to use the following rules:

• The antiderivative of $e^{ax}$ is $\frac{1}{a} e^{ax}$
• The antiderivative of $c$ is $cx$, where $c$ is a constant
• The antiderivative of $x^n$ is $\frac{x^{n+1}}{n+1}$

Q: What is the difference between a definite integral and an indefinite integral?

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A: A definite integral is a mathematical concept that represents the area under a curve between two points, while an indefinite integral is a mathematical concept that represents the antiderivative of a function. The definite integral is denoted by $\int_a^b f(x) d x$, while the indefinite integral is denoted by $\int f(x) d x$.

Q: Can I use a calculator to evaluate a definite integral?

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A: Yes, you can use a calculator to evaluate a definite integral. Most calculators have a built-in function to evaluate definite integrals. However, it's always a good idea to check your work by hand to make sure you get the correct answer.

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

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

• Forgetting to evaluate the antiderivative at the limits of integration
• Forgetting to subtract the value of the antiderivative at the lower limit of integration from the value of the antiderivative at the upper limit of integration
• Not using the correct rules to find the antiderivative of a function

Conclusion

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In this article, we answered some frequently asked questions related to evaluating definite integrals. We hope this article has been helpful in clarifying some of the concepts related to definite integrals.

Final Answer

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The final answer is $\boxed{-50 e^{-1} + 35}$.

References

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• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart

Tags

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• Calculus
• Definite Integral
• Antiderivative
• Fundamental Theorem of Calculus
• Exponential Function
• Trigonometric Function