Evaluate ∫ 0 20 ( 9 E 0.25 X + 2 ) D X \int_0^{20}\left(9 E^{0.25 X}+2\right) D X ∫ 0 20 ​ ( 9 E 0.25 X + 2 ) D X And Express The Answer In Simplest Form.

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Introduction

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In mathematics, definite integrals are a fundamental concept used to calculate the area under curves and solve various problems in physics, engineering, and other fields. In this article, we will evaluate the definite integral $\int_0^{20}\left(9 e^{0.25 x}+2\right) d x$ and express the answer in simplest form.

Understanding Definite Integrals

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A definite integral is a mathematical expression that represents the area under a curve between two points. It is denoted by the symbol $\int_a^b f(x) dx$, where $f(x)$ is the function being integrated, and $a$ and $b$ are the limits of integration.

To evaluate a definite integral, we need to follow the following steps:

1. Integrate the function: Find the antiderivative of the function $f(x)$.
2. Apply the Fundamental Theorem of Calculus: Use the Fundamental Theorem of Calculus to evaluate the definite integral.
3. Simplify the expression: Simplify the resulting expression to obtain the final answer.

Evaluating the Definite Integral

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To evaluate the definite integral $\int_0^{20}\left(9 e^{0.25 x}+2\right) d x$, we need to follow the steps outlined above.

Step 1: Integrate the function

The first step is to integrate the function $f(x) = 9 e^{0.25 x} + 2$. To do this, we need to find the antiderivative of the function.

The antiderivative of $e^{0.25 x}$ is $\frac{1}{0.25} e^{0.25 x} = 4 e^{0.25 x}$.

The antiderivative of $2$ is $2x$.

Therefore, the antiderivative of $f(x)$ is $4 e^{0.25 x} + 2x$.

Step 2: Apply the Fundamental Theorem of Calculus

The next step is to apply the Fundamental Theorem of Calculus to evaluate the definite integral.

The Fundamental Theorem of Calculus states that $\int_a^b f(x) dx = F(b) - F(a)$, where $F(x)$ is the antiderivative of $f(x)$.

In this case, we have:

$\int_0^{20}\left(9 e^{0.25 x}+2\right) d x = \left[4 e^{0.25 x} + 2x\right]_0^{20}$

Evaluating the expression at the limits of integration, we get:

$\left[4 e^{0.25 x} + 2x\right]_0^{20} = \left(4 e^{0.25 \cdot 20} + 2 \cdot 20\right) - \left(4 e^{0.25 \cdot 0} + 2 \cdot 0\right)$

Simplifying the expression, we get:

$\left(4 e^{5} + 40\right) - \left(4 e^{0} + 0\right)$

Since $e^0 = 1$, we can simplify the expression further:

$\left(4 e^{5} + 40\right) - 4$

Step 3: Simplify the expression

The final step is to simplify the expression to obtain the final answer.

We can simplify the expression by combining the terms:

$4 e^{5} + 40 - 4 = 4 e^{5} + 36$

Therefore, the final answer is:

$\boxed{4 e^{5} + 36}$

Conclusion

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In this article, we evaluated the definite integral $\int_0^{20}\left(9 e^{0.25 x}+2\right) d x$ and expressed the answer in simplest form. We followed the steps outlined above to integrate the function, apply the Fundamental Theorem of Calculus, and simplify the expression to obtain the final answer.

The final answer is $\boxed{4 e^{5} + 36}$.

References

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• [1] Calculus by Michael Spivak. Publish or Perish, Inc., 2008.
• [2] Calculus: Early Transcendentals by James Stewart. Cengage Learning, 2016.

Glossary

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• Definite integral: A mathematical expression that represents the area under a curve between two points.
• Antiderivative: A function that is the derivative of another function.
• Fundamental Theorem of Calculus: A theorem that states that the definite integral of a function can be evaluated by finding the antiderivative of the function and applying the Fundamental Theorem of Calculus.

Further Reading

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• Calculus by Michael Spivak. Publish or Perish, Inc., 2008.
• Calculus: Early Transcendentals by James Stewart. Cengage Learning, 2016.

Note: The references and further reading section are for additional information and resources for readers who want to learn more about the topic.

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Introduction

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In our previous article, we evaluated the definite integral $\int_0^{20}\left(9 e^{0.25 x}+2\right) d x$ and expressed the answer in simplest form. In this article, we will answer some frequently asked questions about evaluating definite integrals.

Q&A

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Q: What is a definite integral?

A: A definite integral is a mathematical expression that represents the area under a curve between two points. It is denoted by the symbol $\int_a^b f(x) dx$, where $f(x)$ is the function being integrated, and $a$ and $b$ are the limits of integration.

Q: How do I evaluate a definite integral?

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

1. Integrate the function: Find the antiderivative of the function $f(x)$.
2. Apply the Fundamental Theorem of Calculus: Use the Fundamental Theorem of Calculus to evaluate the definite integral.
3. Simplify the expression: Simplify the resulting expression to obtain the final answer.

Q: What is the Fundamental Theorem of Calculus?

A: The Fundamental Theorem of Calculus is a theorem that states that the definite integral of a function can be evaluated by finding the antiderivative of the function and applying the Fundamental Theorem of Calculus. 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?

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

• Power rule: If $f(x) = x^n$, then $F(x) = \frac{x^{n+1}}{n+1} + C$.
• Exponential rule: If $f(x) = e^x$, then $F(x) = e^x + C$.
• Trigonometric rule: If $f(x) = \sin x$, then $F(x) = -\cos x + C$.

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

A: A definite integral is a mathematical expression that represents the area under a curve between two points, while an indefinite integral is a function that is the antiderivative of another function.

Q: How do I simplify a definite integral?

A: To simplify a definite integral, you need to use the following rules:

• Combine like terms: Combine any like terms in the expression.
• Simplify fractions: Simplify any fractions in the expression.
• Use algebraic manipulations: Use algebraic manipulations to simplify the expression.

Conclusion

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In this article, we answered some frequently asked questions about evaluating definite integrals. We covered topics such as the definition of a definite integral, how to evaluate a definite integral, and how to simplify a definite integral.

References

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• [1] Calculus by Michael Spivak. Publish or Perish, Inc., 2008.
• [2] Calculus: Early Transcendentals by James Stewart. Cengage Learning, 2016.

Glossary

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• Definite integral: A mathematical expression that represents the area under a curve between two points.
• Antiderivative: A function that is the derivative of another function.
• Fundamental Theorem of Calculus: A theorem that states that the definite integral of a function can be evaluated by finding the antiderivative of the function and applying the Fundamental Theorem of Calculus.

Further Reading

---

• Calculus by Michael Spivak. Publish or Perish, Inc., 2008.
• Calculus: Early Transcendentals by James Stewart. Cengage Learning, 2016.

Note: The references and further reading section are for additional information and resources for readers who want to learn more about the topic.