Evaluate ∫ 0 2 ( 10 E 0.5 X + 8 X ) D X \int_0^2\left(10 E^{0.5 X}+8 X\right) D X ∫ 0 2 ​ ( 10 E 0.5 X + 8 X ) 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 $f(x) = 10 e^{0.5 x} + 8 x$ from $x = 0$ to $x = 2$. 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 $10 e^{0.5 x}$

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To find the antiderivative of $10 e^{0.5 x}$, we will use the formula for the antiderivative of an exponential function, which is $\int e^{ax} dx = \frac{1}{a} e^{ax} + C$. In this case, $a = 0.5$, so the antiderivative of $10 e^{0.5 x}$ is $\frac{10}{0.5} e^{0.5 x} + C = 20 e^{0.5 x} + C$.

Step 2: Find the Antiderivative of $8 x$

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To find the antiderivative of $8 x$, we will use the formula for the antiderivative of a polynomial function, which is $\int x^n dx = \frac{x^{n+1}}{n+1} + C$. In this case, $n = 1$, so the antiderivative of $8 x$ is $\frac{8 x^2}{2} + C = 4 x^2 + C$.

Step 3: Evaluate the Definite Integral

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Now that we have found the antiderivatives of $10 e^{0.5 x}$ and $8 x$, we can evaluate the definite integral by applying the fundamental theorem of calculus. The definite integral of $f(x) = 10 e^{0.5 x} + 8 x$ from $x = 0$ to $x = 2$ is equal to the antiderivative of $f(x)$ evaluated at $x = 2$ minus the antiderivative of $f(x)$ evaluated at $x = 0$. Therefore, we have:

$$\int_0^2\left(10 e^{0.5 x}+8 x\right) d x = \left[20 e^{0.5 x} + 4 x^2\right]_0^2 = \left(20 e^{0.5 \cdot 2} + 4 \cdot 2^2\right) - \left(20 e^{0.5 \cdot 0} + 4 \cdot 0^2\right)$$

Step 4: Simplify the Expression

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To simplify the expression, we will evaluate the exponential functions and the polynomial expressions separately. We have:

$$\left(20 e^{0.5 \cdot 2} + 4 \cdot 2^2\right) - \left(20 e^{0.5 \cdot 0} + 4 \cdot 0^2\right) = \left(20 e^1 + 4 \cdot 4\right) - \left(20 e^0 + 4 \cdot 0\right)$$

Since $e^0 = 1$, we have:

$$\left(20 e^1 + 4 \cdot 4\right) - \left(20 e^0 + 4 \cdot 0\right) = \left(20 e^1 + 16\right) - 20$$

Step 5: Final Answer

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To find the final answer, we will simplify the expression further. We have:

$$\left(20 e^1 + 16\right) - 20 = 20 e^1 - 4$$

Since $e^1 \approx 2.71828$, we have:

$$20 e^1 - 4 \approx 20 \cdot 2.71828 - 4 \approx 54.3656 - 4 \approx 50.3656$$

However, we are asked to express the answer in simplest form, so we will leave the answer in terms of $e$. Therefore, the final answer is:

$$20 e^1 - 4$$

Conclusion

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In this article, we evaluated the definite integral of the function $f(x) = 10 e^{0.5 x} + 8 x$ from $x = 0$ to $x = 2$ and expressed the answer in simplest form. We used the fundamental theorem of calculus to evaluate the definite integral and simplified the expression to obtain the final answer. The final answer is $20 e^1 - 4$, which is the simplest form of the answer.

References

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

Keywords

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• Definite integral
• Fundamental theorem of calculus
• Antiderivative
• Exponential function
• Polynomial function
• Calculus
• Mathematics
  

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Introduction

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In our previous article, we evaluated the definite integral of the function $f(x) = 10 e^{0.5 x} + 8 x$ from $x = 0$ to $x = 2$ and expressed the answer in simplest form. In this article, we will answer some frequently asked questions about 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 the symbol $\int_a^b f(x) dx$ and is used to calculate 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 the lower limit of integration from the value of the antiderivative at the upper limit of integration.

Q: What is the fundamental theorem of calculus?

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A: The fundamental theorem of calculus is a mathematical theorem 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 the symbol $\int_a^b f(x) dx = F(b) - F(a)$, 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 $x^n$ is $\frac{x^{n+1}}{n+1}$.
• The antiderivative of $e^{ax}$ is $\frac{1}{a} e^{ax}$.
• The antiderivative of $\sin(ax)$ is $-\frac{1}{a} \cos(ax)$.
• The antiderivative of $\cos(ax)$ is $\frac{1}{a} \sin(ax)$.

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 the symbol $\int_a^b f(x) dx$, while the indefinite integral is denoted by the symbol $\int f(x) dx$.

Q: How do I use the fundamental theorem of calculus to evaluate a definite integral?

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A: To use the fundamental theorem of calculus 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 the lower limit of integration from the value of the antiderivative at the upper limit of integration.

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 upper and lower 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.
• Using the wrong antiderivative or evaluating the antiderivative at the wrong limits of integration.

Conclusion

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In this article, we answered some frequently asked questions about evaluating definite integrals. We discussed the fundamental theorem of calculus, the antiderivative, and some common mistakes to avoid when evaluating definite integrals. We hope that this article has been helpful in clarifying some of the concepts and procedures involved in evaluating definite integrals.

References

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

Keywords

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• Definite integral
• Fundamental theorem of calculus
• Antiderivative
• Exponential function
• Polynomial function
• Calculus
• Mathematics