Evaluate The Integral:$\int -\frac{1}{8^x} \, Dx$

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Introduction

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In this article, we will evaluate the integral of a negative exponential function, specifically the integral $\int -\frac{1}{8^x} \, dx$. This type of integral is commonly encountered in calculus and is an essential part of understanding the properties of exponential functions.

Understanding the Integral

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The given integral is $\int -\frac{1}{8^x} \, dx$. To evaluate this integral, we need to understand the properties of the exponential function and how to integrate it. The exponential function $a^x$ is defined as $e^{x\ln(a)}$, where $e$ is the base of the natural logarithm and $\ln(a)$ is the natural logarithm of $a$. In this case, we have $a = 8$, so the integral becomes $\int -\frac{1}{e^{x\ln(8)}} \, dx$.

Evaluating the Integral

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To evaluate the integral, we can use the property of logarithms that states $\ln(a^b) = b\ln(a)$. Applying this property to the integral, we get $\int -\frac{1}{e^{x\ln(8)}} \, dx = \int -\frac{1}{e^{\ln(8^x)}} \, dx$. Now, we can simplify the integral by using the property of exponents that states $e^{\ln(a)} = a$. Applying this property to the integral, we get $\int -\frac{1}{e^{\ln(8^x)}} \, dx = \int -\frac{1}{8^x} \, dx$.

Using the Power Rule of Integration

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The power rule of integration states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$. In this case, we have $n = -1$, so the power rule of integration becomes $\int x^{-1} \, dx = \ln|x| + C$. We can apply this rule to the integral by substituting $x = 8^x$ and $dx = 8^x \ln(8) dx$. This gives us $\int -\frac{1}{8^x} \, dx = -\int \frac{1}{8^x} \, dx = -\ln|8^x| + C$.

Simplifying the Integral

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We can simplify the integral by using the property of logarithms that states $\ln(a^b) = b\ln(a)$. Applying this property to the integral, we get $-\ln|8^x| + C = -x\ln(8) + C$.

Conclusion

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In this article, we evaluated the integral of a negative exponential function, specifically the integral $\int -\frac{1}{8^x} \, dx$. We used the properties of logarithms and exponents to simplify the integral and ultimately arrived at the solution $-x\ln(8) + C$. This type of integral is commonly encountered in calculus and is an essential part of understanding the properties of exponential functions.

Example Problems

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Problem 1

Evaluate the integral $\int -\frac{1}{4^x} \, dx$.

Solution

Using the same method as before, we can evaluate the integral as follows:

$\int -\frac{1}{4^x} \, dx = -\int \frac{1}{4^x} \, dx = -\ln|4^x| + C = -x\ln(4) + C$

Problem 2

Evaluate the integral $\int -\frac{1}{16^x} \, dx$.

Solution

Using the same method as before, we can evaluate the integral as follows:

$\int -\frac{1}{16^x} \, dx = -\int \frac{1}{16^x} \, dx = -\ln|16^x| + C = -x\ln(16) + C$

Applications of the Integral

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The integral $\int -\frac{1}{8^x} \, dx$ has several applications in mathematics and science. For example, it can be used to model population growth and decay, as well as to solve problems involving exponential decay.

Population Growth and Decay

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The integral $\int -\frac{1}{8^x} \, dx$ can be used to model population growth and decay. For example, if we have a population that is growing at a rate of $8^x$ per unit time, the integral can be used to find the total population at any given time.

Exponential Decay

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The integral $\int -\frac{1}{8^x} \, dx$ can also be used to model exponential decay. For example, if we have a substance that is decaying at a rate of $8^x$ per unit time, the integral can be used to find the amount of substance remaining at any given time.

Conclusion

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In this article, we evaluated the integral of a negative exponential function, specifically the integral $\int -\frac{1}{8^x} \, dx$. We used the properties of logarithms and exponents to simplify the integral and ultimately arrived at the solution $-x\ln(8) + C$. This type of integral is commonly encountered in calculus and is an essential part of understanding the properties of exponential functions.

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Q: What is the integral of a negative exponential function?

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A: The integral of a negative exponential function is a mathematical expression that represents the accumulation of a negative exponential function over a given interval. In this article, we evaluated the integral $\int -\frac{1}{8^x} \, dx$.

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

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A: To evaluate the integral of a negative exponential function, you can use the properties of logarithms and exponents. Specifically, you can use the property that $\ln(a^b) = b\ln(a)$ to simplify the integral.

Q: What is the solution to the integral $\int -\frac{1}{8^x} \, dx$?

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A: The solution to the integral $\int -\frac{1}{8^x} \, dx$ is $-x\ln(8) + C$, where $C$ is the constant of integration.

Q: Can I use the power rule of integration to evaluate the integral of a negative exponential function?

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A: Yes, you can use the power rule of integration to evaluate the integral of a negative exponential function. Specifically, you can use the power rule to integrate the function $x^{-1}$, which is equivalent to the negative exponential function $-\frac{1}{8^x}$.

Q: What are some applications of the integral of a negative exponential function?

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A: The integral of a negative exponential function has several applications in mathematics and science. For example, it can be used to model population growth and decay, as well as to solve problems involving exponential decay.

Q: How do I use the integral of a negative exponential function to model population growth and decay?

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A: To use the integral of a negative exponential function to model population growth and decay, you can substitute the population growth or decay rate into the integral and solve for the total population at any given time.

Q: Can I use the integral of a negative exponential function to solve problems involving exponential decay?

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A: Yes, you can use the integral of a negative exponential function to solve problems involving exponential decay. Specifically, you can use the integral to find the amount of substance remaining at any given time.

Q: What is the constant of integration in the solution to the integral $\int -\frac{1}{8^x} \, dx$?

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A: The constant of integration in the solution to the integral $\int -\frac{1}{8^x} \, dx$ is $C$, which is an arbitrary constant that can take on any value.

Q: Can I use the integral of a negative exponential function to solve problems involving exponential growth?

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A: Yes, you can use the integral of a negative exponential function to solve problems involving exponential growth. Specifically, you can use the integral to find the total population at any given time.

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

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A: Some common mistakes to avoid when evaluating the integral of a negative exponential function include:

• Failing to use the properties of logarithms and exponents to simplify the integral
• Using the wrong power rule of integration
• Failing to include the constant of integration in the solution
• Using the integral to solve problems involving exponential growth or decay without properly substituting the population growth or decay rate into the integral.

Q: How do I know if I have correctly evaluated the integral of a negative exponential function?

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A: To know if you have correctly evaluated the integral of a negative exponential function, you can check your work by:

• Verifying that you have used the correct properties of logarithms and exponents to simplify the integral
• Checking that you have used the correct power rule of integration
• Ensuring that you have included the constant of integration in the solution
• Using the integral to solve problems involving exponential growth or decay and verifying that your solution is correct.