Evaluate ∫ 0 2 ( E 0.5 X + 2 X ) D X \int_0^2 \left(e^{0.5 X} + 2x\right) \, Dx ∫ 0 2 ​ ( E 0.5 X + 2 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 $\left(e^{0.5 x} + 2x\right)$ from $0$ to $2$. This involves finding the antiderivative of the function and then applying the fundamental theorem of calculus to obtain the definite integral.

Step 1: Find the Antiderivative of $e^{0.5 x}$

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The antiderivative of $e^{0.5 x}$ is given by $\frac{2}{0.5} e^{0.5 x} = 4e^{0.5 x}$.

Step 2: Find the Antiderivative of $2x$

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The antiderivative of $2x$ is given by $x^2$.

Step 3: Combine the Antiderivatives

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The antiderivative of $\left(e^{0.5 x} + 2x\right)$ is given by $4e^{0.5 x} + x^2$.

Step 4: Apply the Fundamental Theorem of Calculus

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The fundamental theorem of calculus states that the definite integral of a function $f(x)$ from $a$ to $b$ is given by $F(b) - F(a)$, where $F(x)$ is the antiderivative of $f(x)$.

Step 5: Evaluate the Definite Integral

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We need to evaluate the definite integral of $\left(e^{0.5 x} + 2x\right)$ from $0$ to $2$. This involves substituting the values of $x$ into the antiderivative and then subtracting the value of the antiderivative at $x=0$ from the value of the antiderivative at $x=2$.

Step 6: Calculate the Value of the Antiderivative at $x=2$

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The value of the antiderivative at $x=2$ is given by $4e^{0.5(2)} + (2)^2 = 4e^1 + 4$.

Step 7: Calculate the Value of the Antiderivative at $x=0$

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The value of the antiderivative at $x=0$ is given by $4e^{0.5(0)} + (0)^2 = 4e^0 + 0 = 4$.

Step 8: Subtract the Value of the Antiderivative at $x=0$ from the Value of the Antiderivative at $x=2$

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The value of the definite integral is given by $4e^1 + 4 - 4 = 4e^1$.

Step 9: Simplify the Answer

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The value of the definite integral is given by $4e^1 = 4e^1 = 4 \cdot 2.71828 \approx 10.87312$.

Conclusion

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In this article, we evaluated the definite integral of the function $\left(e^{0.5 x} + 2x\right)$ from $0$ to $2$ and expressed the answer in simplest form. The value of the definite integral is given by $4e^1 = 4 \cdot 2.71828 \approx 10.87312$.

Final Answer

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

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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Frequently Asked Questions

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Q: What is the definite integral of $\left(e^{0.5 x} + 2x\right)$ from $0$ to $2$?

A: The definite integral of $\left(e^{0.5 x} + 2x\right)$ from $0$ to $2$ is given by $4e^1 = 4 \cdot 2.71828 \approx 10.87312$.

Q: How do I find the antiderivative of $e^{0.5 x}$?

A: The antiderivative of $e^{0.5 x}$ is given by $\frac{2}{0.5} e^{0.5 x} = 4e^{0.5 x}$.

Q: How do I find the antiderivative of $2x$?

A: The antiderivative of $2x$ is given by $x^2$.

Q: How do I combine the antiderivatives of $e^{0.5 x}$ and $2x$?

A: The antiderivative of $\left(e^{0.5 x} + 2x\right)$ is given by $4e^{0.5 x} + x^2$.

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

A: The fundamental theorem of calculus states that the definite integral of a function $f(x)$ from $a$ to $b$ is given by $F(b) - F(a)$, where $F(x)$ is the antiderivative of $f(x)$.

Q: How do I evaluate the definite integral of $\left(e^{0.5 x} + 2x\right)$ from $0$ to $2$?

A: We need to substitute the values of $x$ into the antiderivative and then subtract the value of the antiderivative at $x=0$ from the value of the antiderivative at $x=2$.

Q: What is the value of the antiderivative at $x=2$?

A: The value of the antiderivative at $x=2$ is given by $4e^{0.5(2)} + (2)^2 = 4e^1 + 4$.

Q: What is the value of the antiderivative at $x=0$?

A: The value of the antiderivative at $x=0$ is given by $4e^{0.5(0)} + (0)^2 = 4e^0 + 0 = 4$.

Q: How do I simplify the answer?

A: The value of the definite integral is given by $4e^1 = 4 \cdot 2.71828 \approx 10.87312$.

Additional Resources

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• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart
• [3] Khan Academy: Calculus
• [4] MIT OpenCourseWare: Calculus

Tags

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• Calculus
• Definite Integral
• Antiderivative
• Fundamental Theorem of Calculus
• Exponential Function
• Trigonometric Function
• Math
• Science
• Education