Evaluate The Integral: $ \int 4^{-x} , Dx $

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Introduction

In mathematics, integration is a fundamental concept that plays a crucial role in solving problems in various fields, including physics, engineering, and economics. One of the most common types of integrals is the power integral, which involves integrating a function of the form $x^n$. However, there are cases where the integral may not be in the standard form, and we need to manipulate it to evaluate it. In this article, we will discuss how to evaluate the integral of $4^{-x}$, which is a classic example of a non-standard integral.

Understanding the Integral

The integral we are dealing with is $ \int 4^{-x} , dx $. To evaluate this integral, we need to understand the properties of the function $4^{-x}$. This function can be rewritten as $ \frac{1}{4^x} $, which is a reciprocal function. The reciprocal function is a type of function that has a constant denominator, and its integral is not straightforward to evaluate.

Manipulating the Integral

To evaluate the integral of $4^{-x}$, we need to manipulate it to a form that we can integrate. One way to do this is to use the property of exponents that states $a^{-x} = \frac{1}{a^x}$. Using this property, we can rewrite the integral as $ \int \frac{1}{4^x} , dx $. Now, we can see that the integral is in the form of a reciprocal function, which is a type of function that has a constant denominator.

Using the Logarithmic Function

To evaluate the integral of $ \frac{1}{4^x} $, we can use the logarithmic function. The logarithmic function is a function that is the inverse of the exponential function. In this case, we can use the natural logarithm, which is denoted by $ \ln(x) $. The natural logarithm is a function that is defined as the integral of $ \frac{1}{x} $.

Evaluating the Integral

Using the property of logarithms that states $ \ln(a^x) = x \ln(a) $, we can rewrite the integral as $ \int \frac{1}{4^x} , dx = \int 4^{-x} , dx = \int e^{-x \ln(4)} , dx $. Now, we can see that the integral is in the form of an exponential function, which is a type of function that has a constant base.

Applying the Exponential Function Rule

To evaluate the integral of $ e^{-x \ln(4)} $, we can use the exponential function rule, which states that the integral of $ e^{ax} $ is $ \frac{1}{a} e^{ax} $. In this case, we have $ a = -\ln(4) $, so the integral is $ \frac{1}{-\ln(4)} e^{-x \ln(4)} $.

Simplifying the Result

Using the property of exponents that states $ e^{ax} = (e^a)x $, we can rewrite the result as $ \frac{1}{-\ln(4)} (e^{-\ln(4)})x $. Now, we can see that the result is in the form of a constant times a power function.

Conclusion

In this article, we discussed how to evaluate the integral of $4^{-x}$. We used the property of exponents to rewrite the integral as a reciprocal function, and then used the logarithmic function to evaluate the integral. We also used the exponential function rule to simplify the result. The final result is $ \frac{1}{-\ln(4)} (e^{-\ln(4)})x $, which is a constant times a power function.

Final Answer

The final answer to the integral of $4^{-x}$ is $ \frac{1}{-\ln(4)} (e^{-\ln(4)})x $.

Example Problems

Here are some example problems that involve evaluating the integral of $4^{-x}$:

• Evaluate the integral of $4^{-x}$ from $x = 0$ to $x = 1$.
• Evaluate the integral of $4^{-x}$ from $x = 1$ to $x = 2$.
• Evaluate the integral of $4^{-x}$ from $x = 2$ to $x = 3$.

Solutions

Here are the solutions to the example problems:

• Evaluate the integral of $4^{-x}$ from $x = 0$ to $x = 1$:

  $ \int_0^1 4^{-x} , dx = \left[ \frac{1}{-\ln(4)} (e^{-\ln(4)})x \right]_0^1 = \frac{1}{-\ln(4)} (e^{-\ln(4)})1 - \frac{1}{-\ln(4)} (e^{-\ln(4)})0 = \frac{1}{-\ln(4)} e^{-\ln(4)} - \frac{1}{-\ln(4)} = \frac{1}{-\ln(4)} (e^{-\ln(4)} - 1) $

• Evaluate the integral of $4^{-x}$ from $x = 1$ to $x = 2$:

  $ \int_1^2 4^{-x} , dx = \left[ \frac{1}{-\ln(4)} (e^{-\ln(4)})x \right]_1^2 = \frac{1}{-\ln(4)} (e^{-\ln(4)})2 - \frac{1}{-\ln(4)} (e^{-\ln(4)})1 = \frac{1}{-\ln(4)} e^{-2\ln(4)} - \frac{1}{-\ln(4)} e^{-\ln(4)} = \frac{1}{-\ln(4)} (e^{-2\ln(4)} - e^{-\ln(4)}) $

• Evaluate the integral of $4^{-x}$ from $x = 2$ to $x = 3$:

  $ \int_2^3 4^{-x} , dx = \left[ \frac{1}{-\ln(4)} (e^{-\ln(4)})x \right]_2^3 = \frac{1}{-\ln(4)} (e^{-\ln(4)})3 - \frac{1}{-\ln(4)} (e^{-\ln(4)})2 = \frac{1}{-\ln(4)} e^{-3\ln(4)} - \frac{1}{-\ln(4)} e^{-2\ln(4)} = \frac{1}{-\ln(4)} (e^{-3\ln(4)} - e^{-2\ln(4)}) $

Conclusion

In this article, we discussed how to evaluate the integral of $4^{-x}$. We used the property of exponents to rewrite the integral as a reciprocal function, and then used the logarithmic function to evaluate the integral. We also used the exponential function rule to simplify the result. The final result is $ \frac{1}{-\ln(4)} (e^{-\ln(4)})x $, which is a constant times a power function. We also provided example problems and solutions to help illustrate the concept.

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Introduction

In mathematics, integration is a fundamental concept that plays a crucial role in solving problems in various fields, including physics, engineering, and economics. One of the most common types of integrals is the power integral, which involves integrating a function of the form $x^n$. However, there are cases where the integral may not be in the standard form, and we need to manipulate it to evaluate it. In this article, we will discuss how to evaluate the integral of $4^{-x}$, which is a classic example of a non-standard integral.

Understanding the Integral

The integral we are dealing with is $ \int 4^{-x} , dx $. To evaluate this integral, we need to understand the properties of the function $4^{-x}$. This function can be rewritten as $ \frac{1}{4^x} $, which is a reciprocal function. The reciprocal function is a type of function that has a constant denominator, and its integral is not straightforward to evaluate.

Manipulating the Integral

To evaluate the integral of $4^{-x}$, we need to manipulate it to a form that we can integrate. One way to do this is to use the property of exponents that states $a^{-x} = \frac{1}{a^x}$. Using this property, we can rewrite the integral as $ \int \frac{1}{4^x} , dx $. Now, we can see that the integral is in the form of a reciprocal function, which is a type of function that has a constant denominator.

Using the Logarithmic Function

To evaluate the integral of $ \frac{1}{4^x} $, we can use the logarithmic function. The logarithmic function is a function that is the inverse of the exponential function. In this case, we can use the natural logarithm, which is denoted by $ \ln(x) $. The natural logarithm is a function that is defined as the integral of $ \frac{1}{x} $.

Evaluating the Integral

Using the property of logarithms that states $ \ln(a^x) = x \ln(a) $, we can rewrite the integral as $ \int \frac{1}{4^x} , dx = \int 4^{-x} , dx = \int e^{-x \ln(4)} , dx $. Now, we can see that the integral is in the form of an exponential function, which is a type of function that has a constant base.

Applying the Exponential Function Rule

To evaluate the integral of $ e^{-x \ln(4)} $, we can use the exponential function rule, which states that the integral of $ e^{ax} $ is $ \frac{1}{a} e^{ax} $. In this case, we have $ a = -\ln(4) $, so the integral is $ \frac{1}{-\ln(4)} e^{-x \ln(4)} $.

Simplifying the Result

Using the property of exponents that states $ e^{ax} = (e^a)x $, we can rewrite the result as $ \frac{1}{-\ln(4)} (e^{-\ln(4)})x $. Now, we can see that the result is in the form of a constant times a power function.

Conclusion

In this article, we discussed how to evaluate the integral of $4^{-x}$. We used the property of exponents to rewrite the integral as a reciprocal function, and then used the logarithmic function to evaluate the integral. We also used the exponential function rule to simplify the result. The final result is $ \frac{1}{-\ln(4)} (e^{-\ln(4)})x $, which is a constant times a power function.

Q&A

Q: What is the integral of $4^{-x}$?

A: The integral of $4^{-x}$ is $ \frac{1}{-\ln(4)} (e^{-\ln(4)})x $.

Q: How do I evaluate the integral of $4^{-x}$?

A: To evaluate the integral of $4^{-x}$, you need to use the property of exponents to rewrite the integral as a reciprocal function, and then use the logarithmic function to evaluate the integral.

Q: What is the property of exponents that I need to use to evaluate the integral of $4^{-x}$?

A: The property of exponents that you need to use is $a^{-x} = \frac{1}{a^x}$.

Q: What is the logarithmic function that I need to use to evaluate the integral of $4^{-x}$?

A: The logarithmic function that you need to use is the natural logarithm, which is denoted by $ \ln(x) $.

Q: How do I use the exponential function rule to simplify the result of the integral of $4^{-x}$?

A: To use the exponential function rule, you need to rewrite the result as $ \frac{1}{-\ln(4)} (e^{-\ln(4)})x $, and then use the property of exponents that states $ e^{ax} = (e^a)x $.

Q: What is the final result of the integral of $4^{-x}$?

A: The final result of the integral of $4^{-x}$ is $ \frac{1}{-\ln(4)} (e^{-\ln(4)})x $.

Q: Can I use the integral of $4^{-x}$ to solve any other problems?

A: Yes, you can use the integral of $4^{-x}$ to solve any other problems that involve integrating a function of the form $4^{-x}$.

Q: How do I apply the integral of $4^{-x}$ to solve a problem?

A: To apply the integral of $4^{-x}$ to solve a problem, you need to use the property of exponents to rewrite the integral as a reciprocal function, and then use the logarithmic function to evaluate the integral.

Q: What are some example problems that involve the integral of $4^{-x}$?

A: Some example problems that involve the integral of $4^{-x}$ include evaluating the integral of $4^{-x}$ from $x = 0$ to $x = 1$, evaluating the integral of $4^{-x}$ from $x = 1$ to $x = 2$, and evaluating the integral of $4^{-x}$ from $x = 2$ to $x = 3$.

Q: How do I solve the example problems that involve the integral of $4^{-x}$?

A: To solve the example problems that involve the integral of $4^{-x}$, you need to use the property of exponents to rewrite the integral as a reciprocal function, and then use the logarithmic function to evaluate the integral.

Q: What is the solution to the example problem of evaluating the integral of $4^{-x}$ from $x = 0$ to $x = 1$?

A: The solution to the example problem of evaluating the integral of $4^{-x}$ from $x = 0$ to $x = 1$ is $ \frac{1}{-\ln(4)} (e^{-\ln(4)})1 - \frac{1}{-\ln(4)} (e^{-\ln(4)})0 = \frac{1}{-\ln(4)} e^{-\ln(4)} - \frac{1}{-\ln(4)} = \frac{1}{-\ln(4)} (e^{-\ln(4)} - 1) $.

Q: What is the solution to the example problem of evaluating the integral of $4^{-x}$ from $x = 1$ to $x = 2$?

A: The solution to the example problem of evaluating the integral of $4^{-x}$ from $x = 1$ to $x = 2$ is $ \frac{1}{-\ln(4)} (e^{-\ln(4)})2 - \frac{1}{-\ln(4)} (e^{-\ln(4)})1 = \frac{1}{-\ln(4)} e^{-2\ln(4)} - \frac{1}{-\ln(4)} e^{-\ln(4)} = \frac{1}{-\ln(4)} (e^{-2\ln(4)} - e^{-\ln(4)}) $.

Q: What is the solution to the example problem of evaluating the integral of $4^{-x}$ from $x = 2$ to $x = 3$?

A: The solution to the example problem of evaluating the integral of $4^{-x}$ from $x = 2$ to $x = 3$ is $ \frac{1}{-\ln(4)} (e^{-\ln(4)})3 - \frac{1}{-\ln(4)} (e^{-\ln(4)})2 = \frac{1}{-\ln(4)} e^{-3\ln(4)} - \frac{1}{-\ln(4)} e^{-2\ln(4)} = \frac{1}{-\ln(4)} (e^{-3\ln(4)} - e^{-2\ln(4)}) $.

Conclusion

In this article, we discussed how to evaluate the integral