Evaluate ∫ 0 2 ( 10 E 0.5 X + 8 X ) D X \int_0^2 \left(10 E^{0.5 X} + 8 X\right) \, Dx ∫ 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 such as physics, engineering, and economics.

The Definite Integral

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The definite integral of a function $f(x)$ from $x = a$ to $x = b$ is denoted by $\int_a^b f(x) \, dx$ and is defined as the limit of the sum of the areas of the rectangles that approximate the region under the curve of $f(x)$.

Evaluating the Definite Integral

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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 $x = a$ to $x = b$ is equal to $F(b) - F(a)$, where $F(x)$ is the antiderivative of $f(x)$.

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

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The antiderivative of $f(x) = 10 e^{0.5 x} + 8 x$ is given by:

$$F(x) = \int f(x) \, dx = \int (10 e^{0.5 x} + 8 x) \, dx$$

Using the linearity property of integration, we can write:

$$F(x) = \int 10 e^{0.5 x} \, dx + \int 8 x \, dx$$

Step 2: Evaluate the First Integral

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The first integral is:

$$\int 10 e^{0.5 x} \, dx = 10 \int e^{0.5 x} \, dx$$

Using the substitution method, we can write:

$$u = 0.5 x \Rightarrow du = 0.5 dx$$

$$\int e^{0.5 x} \, dx = \frac{1}{0.5} e^{0.5 x} + C = 2 e^{0.5 x} + C$$

Therefore, the first integral is:

$$\int 10 e^{0.5 x} \, dx = 10 (2 e^{0.5 x} + C) = 20 e^{0.5 x} + C$$

Step 3: Evaluate the Second Integral

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The second integral is:

$$\int 8 x \, dx = 4 x^2 + C$$

Step 4: Combine the Results

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Combining the results of the two integrals, we get:

$$F(x) = 20 e^{0.5 x} + 4 x^2 + C$$

Step 5: Evaluate $F(2)$ and $F(0)$

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Evaluating $F(2)$ and $F(0)$, we get:

$$F(2) = 20 e^{0.5 (2)} + 4 (2)^2 = 20 e^1 + 16$$

$$F(0) = 20 e^{0.5 (0)} + 4 (0)^2 = 20 + 0 = 20$$

Step 6: Find the Definite Integral

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Using the fundamental theorem of calculus, we can write:

$$\int_0^2 f(x) \, dx = F(2) - F(0) = (20 e^1 + 16) - 20 = 20 e^1 - 4$$

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 $x = 2$. The definite integral is equal to $20 e^1 - 4$.

Final Answer

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The final answer is $\boxed{20 e^1 - 4}$.

References

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

Tags

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• definite integral
• antiderivative
• fundamental theorem of calculus
• linearity property of integration
• substitution method
• definite integral of exponential function
• definite integral of polynomial function
  

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Introduction

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In the previous article, we evaluated the definite integral of the function $f(x) = 10 e^{0.5 x} + 8 x$ from $x = 0$ to $x = 2$. In this article, we will answer some frequently asked questions related to the definite integral of exponential and polynomial functions.

Q&A

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Q: What is the definite integral of $e^x$ from $x = 0$ to $x = 1$?

A: The definite integral of $e^x$ from $x = 0$ to $x = 1$ is given by:

$$\int_0^1 e^x \, dx = e^x \Big|_0^1 = e^1 - e^0 = e - 1$$

Q: What is the definite integral of $x^2$ from $x = 0$ to $x = 2$?

A: The definite integral of $x^2$ from $x = 0$ to $x = 2$ is given by:

$$\int_0^2 x^2 \, dx = \frac{x^3}{3} \Big|_0^2 = \frac{2^3}{3} - \frac{0^3}{3} = \frac{8}{3}$$

Q: How do I evaluate the definite integral of a function that has a constant multiple?

A: To evaluate the definite integral of a function that has a constant multiple, you can use the linearity property of integration. For example, if you want to evaluate the definite integral of $3x^2$ from $x = 0$ to $x = 2$, you can write:

$$\int_0^2 3x^2 \, dx = 3 \int_0^2 x^2 \, dx = 3 \left( \frac{x^3}{3} \Big|_0^2 \right) = 3 \left( \frac{2^3}{3} - \frac{0^3}{3} \right) = 3 \left( \frac{8}{3} \right) = 8$$

Q: How do I evaluate the definite integral of a function that has a trigonometric function?

A: To evaluate the definite integral of a function that has a trigonometric function, you can use the substitution method. For example, if you want to evaluate the definite integral of $\sin x$ from $x = 0$ to $x = \pi$, you can write:

$$\int_0^\pi \sin x \, dx = -\cos x \Big|_0^\pi = -\cos \pi + \cos 0 = -(-1) + 1 = 2$$

Q: What is the definite integral of $e^{2x}$ from $x = 0$ to $x = 1$?

A: The definite integral of $e^{2x}$ from $x = 0$ to $x = 1$ is given by:

$$\int_0^1 e^{2x} \, dx = \frac{1}{2} e^{2x} \Big|_0^1 = \frac{1}{2} (e^2 - e^0) = \frac{1}{2} (e^2 - 1)$$

Conclusion

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In this article, we answered some frequently asked questions related to the definite integral of exponential and polynomial functions. We hope that this article has been helpful in understanding the concept of definite integral and how to evaluate it.

Final Answer

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The final answer is $\boxed{2}$.

References

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

Tags

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• definite integral
• exponential function
• polynomial function
• linearity property of integration
• substitution method
• definite integral of trigonometric function