Evaluate The Integral: $\int 3 E^{-(2x+7)} \, Dx$

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Introduction

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In calculus, integrals are used to find the area under curves and are a fundamental concept in mathematics. When dealing with exponential functions, evaluating integrals can be a bit challenging. In this article, we will focus on evaluating the integral of the function $3 e^{-(2x+7)}$ with respect to $x$. We will break down the solution into manageable steps and provide a clear explanation of each step.

Understanding Exponential Functions

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Exponential functions are a type of function that has the form $f(x) = a^x$, where $a$ is a positive constant. In the given integral, the function is $3 e^{-(2x+7)}$, which is an exponential function with base $e$ (approximately 2.718). The exponent is a linear function of $x$, which is $-(2x+7)$.

The Integral of Exponential Functions

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The integral of an exponential function with base $e$ is given by:

$$\int e^x \, dx = e^x + C$$

where $C$ is the constant of integration. However, in the given integral, the exponent is not simply $x$, but rather $-(2x+7)$. To evaluate this integral, we need to use the chain rule of integration.

Using the Chain Rule of Integration

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The chain rule of integration states that if we have an integral of the form:

$$\int f(g(x)) \, dx$$

then we can use the chain rule to evaluate it as:

$$\int f(g(x)) \, dx = F(g(x)) + C$$

where $F$ is the antiderivative of $f$. In our case, we have:

$$\int 3 e^{-(2x+7)} \, dx$$

We can rewrite the exponent as:

$$-(2x+7) = -2x - 7$$

So, we have:

$$\int 3 e^{-2x - 7} \, dx$$

Evaluating the Integral

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To evaluate this integral, we can use the chain rule of integration. We can rewrite the integral as:

$$\int 3 e^{-2x} e^{-7} \, dx$$

Using the property of exponents that states $e^{a+b} = e^a e^b$, we can rewrite the integral as:

$$\int 3 e^{-2x} e^{-7} \, dx = \int 3 e^{-2x} \cdot e^{-7} \, dx$$

Now, we can use the chain rule of integration to evaluate the integral:

$$\int 3 e^{-2x} \cdot e^{-7} \, dx = 3 e^{-7} \int e^{-2x} \, dx$$

Using the formula for the integral of an exponential function, we have:

$$3 e^{-7} \int e^{-2x} \, dx = 3 e^{-7} \left( -\frac{1}{2} e^{-2x} \right) + C$$

Simplifying the expression, we get:

$$3 e^{-7} \left( -\frac{1}{2} e^{-2x} \right) + C = -\frac{3}{2} e^{-7} e^{-2x} + C$$

Simplifying the Expression

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We can simplify the expression further by combining the exponents:

$$-\frac{3}{2} e^{-7} e^{-2x} + C = -\frac{3}{2} e^{-7 - 2x} + C$$

Conclusion

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In this article, we evaluated the integral of the function $3 e^{-(2x+7)}$ with respect to $x$. We used the chain rule of integration to evaluate the integral and simplified the expression to get the final answer. The integral of the function is:

$$-\frac{3}{2} e^{-7 - 2x} + C$$

where $C$ is the constant of integration.

Applications of the Integral

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The integral of the function $3 e^{-(2x+7)}$ has many applications in mathematics and physics. For example, it can be used to model population growth, chemical reactions, and electrical circuits. It can also be used to solve problems in engineering, economics, and computer science.

Future Work

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In the future, we can use the integral of the function $3 e^{-(2x+7)}$ to solve more complex problems. We can also use it to derive new formulas and theorems in mathematics. Additionally, we can use it to model real-world phenomena and make predictions about future events.

References

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• [1] "Calculus" by Michael Spivak
• [2] "Introduction to Calculus" by Michael Sullivan
• [3] "Calculus: Early Transcendentals" by James Stewart

Glossary

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• Exponential function: A function of the form $f(x) = a^x$, where $a$ is a positive constant.
• Chain rule of integration: A rule that states that if we have an integral of the form $\int f(g(x)) \, dx$, then we can use the chain rule to evaluate it as $\int f(g(x)) \, dx = F(g(x)) + C$, where $F$ is the antiderivative of $f$.
• Constant of integration: A constant that is added to the integral to make it equal to the original function.
• Antiderivative: A function that is the integral of another function.
  

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Introduction

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In our previous article, we evaluated the integral of the function $3 e^{-(2x+7)}$ with respect to $x$. We used the chain rule of integration to evaluate the integral and simplified the expression to get the final answer. In this article, we will provide a Q&A guide to help you understand the concept of evaluating integrals of exponential functions.

Q: What is an exponential function?

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A: An exponential function is a function of the form $f(x) = a^x$, where $a$ is a positive constant. Examples of exponential functions include $2^x$, $e^x$, and $3^x$.

Q: What is the chain rule of integration?

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A: The chain rule of integration is a rule that states that if we have an integral of the form $\int f(g(x)) \, dx$, then we can use the chain rule to evaluate it as $\int f(g(x)) \, dx = F(g(x)) + C$, where $F$ is the antiderivative of $f$.

Q: How do I evaluate the integral of an exponential function?

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A: To evaluate the integral of an exponential function, you can use the chain rule of integration. First, rewrite the integral in the form $\int f(g(x)) \, dx$. Then, use the chain rule to evaluate the integral as $\int f(g(x)) \, dx = F(g(x)) + C$, where $F$ is the antiderivative of $f$.

Q: What is the antiderivative of an exponential function?

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A: The antiderivative of an exponential function is a function that is the integral of the exponential function. For example, the antiderivative of $e^x$ is $e^x$ itself.

Q: How do I simplify the expression after evaluating the integral?

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A: To simplify the expression after evaluating the integral, you can combine the exponents and simplify the expression. For example, if you have the expression $e^{-7 - 2x}$, you can simplify it to $e^{-7} e^{-2x}$.

Q: What are some common applications of the integral of exponential functions?

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A: The integral of exponential functions has many applications in mathematics and physics. Some common applications include:

• Modeling population growth
• Chemical reactions
• Electrical circuits
• Engineering
• Economics
• Computer science

Q: Can I use the integral of exponential functions to solve more complex problems?

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A: Yes, you can use the integral of exponential functions to solve more complex problems. For example, you can use it to model real-world phenomena and make predictions about future events.

Q: What are some common mistakes to avoid when evaluating the integral of exponential functions?

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A: Some common mistakes to avoid when evaluating the integral of exponential functions include:

• Not using the chain rule of integration
• Not simplifying the expression after evaluating the integral
• Not combining the exponents
• Not using the correct antiderivative

Q: How can I practice evaluating the integral of exponential functions?

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A: You can practice evaluating the integral of exponential functions by working through examples and exercises. You can also use online resources and calculators to help you evaluate the integral.

Conclusion

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In this article, we provided a Q&A guide to help you understand the concept of evaluating integrals of exponential functions. We covered topics such as exponential functions, the chain rule of integration, antiderivatives, and simplifying expressions. We also discussed common applications of the integral of exponential functions and provided tips for practicing and avoiding common mistakes.

References

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• [1] "Calculus" by Michael Spivak
• [2] "Introduction to Calculus" by Michael Sullivan
• [3] "Calculus: Early Transcendentals" by James Stewart

Glossary

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• Exponential function: A function of the form $f(x) = a^x$, where $a$ is a positive constant.
• Chain rule of integration: A rule that states that if we have an integral of the form $\int f(g(x)) \, dx$, then we can use the chain rule to evaluate it as $\int f(g(x)) \, dx = F(g(x)) + C$, where $F$ is the antiderivative of $f$.
• Antiderivative: A function that is the integral of another function.
• Simplifying expression: Combining the exponents and simplifying the expression after evaluating the integral.