Evaluate The Integral:$\[ \int D U \frac{2 U+3}{4 U+5} = \int D X \\]

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Introduction

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In this article, we will delve into the world of calculus and explore the process of evaluating a specific integral. The integral in question is $\int \frac{2u+3}{4u+5} du = \int dx$. We will break down the steps involved in solving this integral and provide a clear understanding of the underlying concepts.

Understanding the Integral

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Before we begin, it's essential to understand the concept of an integral. An integral is a mathematical operation that calculates the area under a curve or the accumulation of a quantity over a defined interval. In this case, we are dealing with a definite integral, which means we are given a specific interval and are asked to find the area under the curve within that interval.

Step 1: Identify the Integral

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The given integral is $\int \frac{2u+3}{4u+5} du = \int dx$. The first step is to identify the integral and understand what it represents. In this case, we have a rational function of the form $\frac{2u+3}{4u+5}$, and we are asked to find its integral.

Step 2: Choose an Integration Method

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There are several methods for integrating rational functions, including substitution, partial fractions, and integration by parts. In this case, we will use the method of substitution to evaluate the integral.

Step 3: Apply the Substitution Method

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To apply the substitution method, we need to identify a suitable substitution that will simplify the integral. In this case, we can let $v = 4u + 5$. This will allow us to rewrite the integral in terms of $v$ and simplify the expression.

Step 4: Rewrite the Integral in Terms of v

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Using the substitution $v = 4u + 5$, we can rewrite the integral as follows:

$$\int \frac{2u+3}{4u+5} du = \int \frac{2\left(\frac{v-5}{4}\right)+3}{v} dv$$

Step 5: Simplify the Integral

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Now that we have rewritten the integral in terms of $v$, we can simplify the expression. We can start by combining the terms in the numerator:

$$\int \frac{2\left(\frac{v-5}{4}\right)+3}{v} dv = \int \frac{\frac{v-5}{2}+3}{v} dv$$

Step 6: Evaluate the Integral

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Now that we have simplified the integral, we can evaluate it. We can start by multiplying the numerator and denominator by 2 to eliminate the fraction:

$$\int \frac{\frac{v-5}{2}+3}{v} dv = \int \frac{v-5+6}{2v} dv$$

Step 7: Simplify the Integral Further

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We can simplify the integral further by combining the terms in the numerator:

$$\int \frac{v-5+6}{2v} dv = \int \frac{v+1}{2v} dv$$

Step 8: Evaluate the Integral

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Now that we have simplified the integral, we can evaluate it. We can start by using the method of partial fractions to break down the integral into simpler components:

$$\int \frac{v+1}{2v} dv = \frac{1}{2} \int \frac{v}{v} dv + \frac{1}{2} \int \frac{1}{v} dv$$

Step 9: Evaluate the Integral

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Now that we have broken down the integral into simpler components, we can evaluate each component separately. We can start by evaluating the first component:

$$\frac{1}{2} \int \frac{v}{v} dv = \frac{1}{2} \int 1 dv = \frac{1}{2} v$$

Step 10: Evaluate the Second Component

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We can evaluate the second component using the method of logarithmic integration:

$$\frac{1}{2} \int \frac{1}{v} dv = \frac{1}{2} \ln |v| + C$$

Step 11: Combine the Components

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Now that we have evaluated each component, we can combine them to get the final result:

$$\frac{1}{2} v + \frac{1}{2} \ln |v| + C$$

Step 12: Substitute Back to u

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We can substitute back to $u$ by using the original substitution $v = 4u + 5$:

$$\frac{1}{2} (4u + 5) + \frac{1}{2} \ln |4u + 5| + C$$

Conclusion

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In this article, we have evaluated the integral $\int \frac{2u+3}{4u+5} du = \int dx$ using the method of substitution. We have broken down the integral into simpler components and evaluated each component separately. The final result is $\frac{1}{2} (4u + 5) + \frac{1}{2} \ln |4u + 5| + C$. This result provides a clear understanding of the underlying concepts and demonstrates the power of the substitution method in evaluating integrals.

Final Answer

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The final answer is $\boxed{\frac{1}{2} (4u + 5) + \frac{1}{2} \ln |4u + 5| + C}$.

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Introduction

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In our previous article, we evaluated the integral $\int \frac{2u+3}{4u+5} du = \int dx$ using the method of substitution. In this article, we will provide a Q&A section to address any questions or concerns that readers may have.

Q: What is the purpose of the substitution method in evaluating integrals?

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A: The substitution method is a powerful technique used to evaluate integrals by simplifying the expression and making it easier to integrate. In the case of the integral $\int \frac{2u+3}{4u+5} du = \int dx$, we used the substitution $v = 4u + 5$ to simplify the expression and make it easier to integrate.

Q: Why did we choose the substitution $v = 4u + 5$?

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A: We chose the substitution $v = 4u + 5$ because it allowed us to simplify the expression and make it easier to integrate. By letting $v = 4u + 5$, we were able to rewrite the integral in terms of $v$ and simplify the expression.

Q: What is the significance of the constant $C$ in the final answer?

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A: The constant $C$ is an arbitrary constant that represents the family of antiderivatives. In other words, the final answer $\frac{1}{2} (4u + 5) + \frac{1}{2} \ln |4u + 5| + C$ represents a family of antiderivatives, and the constant $C$ can take on any value.

Q: How do we determine the value of the constant $C$?

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A: The value of the constant $C$ is determined by the specific problem or application. In some cases, we may be given a specific value for the constant $C$, while in other cases, we may need to determine the value of the constant $C$ based on the specific problem or application.

Q: What are some common applications of the substitution method in evaluating integrals?

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A: The substitution method is a powerful technique used in a wide range of applications, including physics, engineering, and economics. Some common applications of the substitution method include:

• Evaluating definite integrals
• Finding the area under curves
• Determining the volume of solids
• Solving differential equations

Q: What are some tips for using the substitution method in evaluating integrals?

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A: Here are some tips for using the substitution method in evaluating integrals:

• Choose a substitution that simplifies the expression and makes it easier to integrate.
• Use the substitution to rewrite the integral in terms of the new variable.
• Simplify the expression and make it easier to integrate.
• Use the method of partial fractions to break down the integral into simpler components.
• Evaluate each component separately and combine the results.

Conclusion

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In this article, we have provided a Q&A section to address any questions or concerns that readers may have about evaluating the integral $\int \frac{2u+3}{4u+5} du = \int dx$ using the method of substitution. We hope that this article has been helpful in providing a clear understanding of the underlying concepts and demonstrating the power of the substitution method in evaluating integrals.

Final Answer

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The final answer is $\boxed{\frac{1}{2} (4u + 5) + \frac{1}{2} \ln |4u + 5| + C}$.