Evaluate ∫ 0 4 ( 5 E 0.25 X + 2 X ) D X \int_0^4\left(5 E^{0.25 X}+2 X\right) D X ∫ 0 4 ​ ( 5 E 0.25 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 $f(x) = 5 e^{0.25 x}+2 x$ from $x = 0$ to $x = 4$. 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 $f(x) = 5 e^{0.25 x}+2 x$ is given by:

$$F(x) = \int f(x) dx = \int (5 e^{0.25 x}+2 x) dx$$

Using the linearity property of integration, we can write:

$$F(x) = \int 5 e^{0.25 x} dx + \int 2 x dx$$

The antiderivative of $5 e^{0.25 x}$ is given by:

$$\int 5 e^{0.25 x} dx = 5 \cdot \frac{1}{0.25} e^{0.25 x} + C = 20 e^{0.25 x} + C$$

The antiderivative of $2 x$ is given by:

$$\int 2 x dx = x^2 + C$$

Therefore, the antiderivative of $f(x)$ is given by:

$$F(x) = 20 e^{0.25 x} + x^2 + C$$

Step 2: Evaluate $F(4)$ and $F(0)$

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To evaluate the definite integral, we need to find the values of $F(4)$ and $F(0)$.

$$F(4) = 20 e^{0.25 \cdot 4} + 4^2 + C = 20 e + 16 + C$$

$$F(0) = 20 e^{0.25 \cdot 0} + 0^2 + C = 20 + C$$

Step 3: Find the Definite Integral

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The definite integral is given by:

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

Simplifying the expression, we get:

$$\int_0^4 f(x) dx = 20 e + 16 - 20 = 20 e - 4$$

Conclusion

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In this article, we evaluated the definite integral of the function $f(x) = 5 e^{0.25 x}+2 x$ from $x = 0$ to $x = 4$. The definite integral is given by $\int_0^4 f(x) dx = 20 e - 4$.

Final Answer

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

References

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

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Introduction

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In the previous article, we evaluated the definite integral of the function $f(x) = 5 e^{0.25 x}+2 x$ from $x = 0$ to $x = 4$. In this article, we will answer some frequently asked questions related to evaluating definite integrals.

Q1: What is the difference between a definite integral and an indefinite integral?

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A1: A definite integral is a specific value that is obtained by evaluating an integral over a given interval, whereas an indefinite integral is a general expression that represents the antiderivative of a function.

Q2: How do I evaluate a definite integral?

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A2: To evaluate a definite integral, you need to find the antiderivative of the function and then evaluate it at the upper and lower limits of integration. The definite integral is then given by the difference between the values of the antiderivative at the upper and lower limits.

Q3: What is the fundamental theorem of calculus?

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A3: The fundamental theorem of calculus 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 $f(x)$.

Q4: How do I find the antiderivative of a function?

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A4: To find the antiderivative of a function, you need to use the rules of integration, such as the power rule, the constant multiple rule, and the sum rule. You can also use integration by substitution and integration by parts to find the antiderivative of a function.

Q5: What is the difference between integration by substitution and integration by parts?

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A5: Integration by substitution is a technique used to find the antiderivative of a function by substituting a new variable into the function. Integration by parts is a technique used to find the antiderivative of a function by differentiating one function and integrating the other function.

Q6: How do I use integration by substitution to find the antiderivative of a function?

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A6: To use integration by substitution, you need to identify a new variable that can be substituted into the function. You then need to find the derivative of the new variable and substitute it into the function. The antiderivative of the function is then given by the integral of the function with respect to the new variable.

Q7: How do I use integration by parts to find the antiderivative of a function?

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A7: To use integration by parts, you need to identify two functions that can be differentiated and integrated. You then need to differentiate one function and integrate the other function. The antiderivative of the function is then given by the product of the two functions.

Q8: What is the difference between a definite integral and a definite sum?

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A8: A definite integral is a specific value that is obtained by evaluating an integral over a given interval, whereas a definite sum is a specific value that is obtained by evaluating a sum over a given interval.

Q9: How do I evaluate a definite sum?

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A9: To evaluate a definite sum, you need to find the sum of the terms in the sum and then evaluate it at the upper and lower limits of summation. The definite sum is then given by the difference between the values of the sum at the upper and lower limits.

Q10: What is the difference between a definite integral and a definite product?

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A10: A definite integral is a specific value that is obtained by evaluating an integral over a given interval, whereas a definite product is a specific value that is obtained by evaluating a product over a given interval.

Conclusion

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In this article, we answered some frequently asked questions related to evaluating definite integrals. We hope that this article has provided you with a better understanding of the concepts and techniques involved in evaluating definite integrals.

Final Answer

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

References

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