Evaluate $\int_3^4 \frac{4x^2-15x+16}{x-2} \, Dx$. Write Your Answer In Simplest Form.

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Introduction

In this article, we will evaluate the definite integral of the rational function $\frac{4x^2-15x+16}{x-2}$ from $x=3$ to $x=4$. To do this, we will first factor the numerator and then use partial fraction decomposition to simplify the expression. Finally, we will integrate the resulting expression and evaluate the definite integral.

Factor the Numerator

The numerator of the rational function is $4x^2-15x+16$. We can factor this expression as follows:

$$4x^2-15x+16 = (4x-8)(x-2)$$

Partial Fraction Decomposition

Now that we have factored the numerator, we can use partial fraction decomposition to simplify the expression. We can write the rational function as follows:

$$\frac{4x^2-15x+16}{x-2} = \frac{(4x-8)(x-2)}{x-2}$$

We can cancel out the common factor of $x-2$ to get:

$$\frac{4x^2-15x+16}{x-2} = 4x-8$$

Integrate the Expression

Now that we have simplified the expression, we can integrate it. We can use the power rule of integration to get:

$$\int (4x-8) \, dx = 2x^2-8x+C$$

Evaluate the Definite Integral

Now that we have integrated the expression, we can evaluate the definite integral. We can use the fundamental theorem of calculus to get:

$$\int_3^4 \frac{4x^2-15x+16}{x-2} \, dx = \left[2x^2-8x\right]_3^4$$

We can evaluate the expression at the upper and lower limits of integration to get:

$$\int_3^4 \frac{4x^2-15x+16}{x-2} \, dx = (2(4)^2-8(4))-(2(3)^2-8(3))$$

We can simplify the expression to get:

$$\int_3^4 \frac{4x^2-15x+16}{x-2} \, dx = (32-32)-(18-24)$$

We can further simplify the expression to get:

$$\int_3^4 \frac{4x^2-15x+16}{x-2} \, dx = 0+6$$

We can finally simplify the expression to get:

$$\int_3^4 \frac{4x^2-15x+16}{x-2} \, dx = 6$$

Conclusion

In this article, we evaluated the definite integral of the rational function $\frac{4x^2-15x+16}{x-2}$ from $x=3$ to $x=4$. We first factored the numerator and then used partial fraction decomposition to simplify the expression. Finally, we integrated the resulting expression and evaluated the definite integral. The final answer is $\boxed{6}$.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

1. Factor the numerator: $4x^2-15x+16 = (4x-8)(x-2)$
2. Use partial fraction decomposition: $\frac{4x^2-15x+16}{x-2} = \frac{(4x-8)(x-2)}{x-2}$
3. Cancel out the common factor: $\frac{4x^2-15x+16}{x-2} = 4x-8$
4. Integrate the expression: $\int (4x-8) \, dx = 2x^2-8x+C$
5. Evaluate the definite integral: $\int_3^4 \frac{4x^2-15x+16}{x-2} \, dx = \left[2x^2-8x\right]_3^4$
6. Simplify the expression: $\int_3^4 \frac{4x^2-15x+16}{x-2} \, dx = (32-32)-(18-24)$
7. Further simplify the expression: $\int_3^4 \frac{4x^2-15x+16}{x-2} \, dx = 0+6$
8. Final answer: $\int_3^4 \frac{4x^2-15x+16}{x-2} \, dx = 6$

Frequently Asked Questions

Here are some frequently asked questions about the problem:

• Q: What is the numerator of the rational function? A: The numerator of the rational function is $4x^2-15x+16$.
• Q: How do we factor the numerator? A: We can factor the numerator as $(4x-8)(x-2)$.
• Q: What is the partial fraction decomposition of the rational function? A: The partial fraction decomposition of the rational function is $\frac{(4x-8)(x-2)}{x-2}$.
• Q: How do we integrate the expression? A: We can integrate the expression using the power rule of integration.
• Q: What is the final answer to the problem? A: The final answer to the problem is $\boxed{6}$.

Related Problems

Here are some related problems to the problem:

• Evaluate $\int_2^3 \frac{x^2-4x+3}{x-1} \, dx$.
• Evaluate $\int_1^2 \frac{2x^2-5x+2}{x-1} \, dx$.
• Evaluate $\int_0^1 \frac{x^2-3x+2}{x-1} \, dx$.

Conclusion

In this article, we evaluated the definite integral of the rational function $\frac{4x^2-15x+16}{x-2}$ from $x=3$ to $x=4$. We first factored the numerator and then used partial fraction decomposition to simplify the expression. Finally, we integrated the resulting expression and evaluated the definite integral. The final answer is $\boxed{6}$.

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Introduction

In this article, we will answer some frequently asked questions about the problem of evaluating the definite integral of the rational function $\frac{4x^2-15x+16}{x-2}$ from $x=3$ to $x=4$. We will provide step-by-step solutions to the problem and answer some common questions that students may have.

Q&A

Q: What is the numerator of the rational function?

A: The numerator of the rational function is $4x^2-15x+16$.

Q: How do we factor the numerator?

A: We can factor the numerator as $(4x-8)(x-2)$.

Q: What is the partial fraction decomposition of the rational function?

A: The partial fraction decomposition of the rational function is $\frac{(4x-8)(x-2)}{x-2}$.

Q: How do we integrate the expression?

A: We can integrate the expression using the power rule of integration.

Q: What is the final answer to the problem?

A: The final answer to the problem is $\boxed{6}$.

Q: Can you provide a step-by-step solution to the problem?

A: Yes, here is a step-by-step solution to the problem:

1. Factor the numerator: $4x^2-15x+16 = (4x-8)(x-2)$
2. Use partial fraction decomposition: $\frac{4x^2-15x+16}{x-2} = \frac{(4x-8)(x-2)}{x-2}$
3. Cancel out the common factor: $\frac{4x^2-15x+16}{x-2} = 4x-8$
4. Integrate the expression: $\int (4x-8) \, dx = 2x^2-8x+C$
5. Evaluate the definite integral: $\int_3^4 \frac{4x^2-15x+16}{x-2} \, dx = \left[2x^2-8x\right]_3^4$
6. Simplify the expression: $\int_3^4 \frac{4x^2-15x+16}{x-2} \, dx = (32-32)-(18-24)$
7. Further simplify the expression: $\int_3^4 \frac{4x^2-15x+16}{x-2} \, dx = 0+6$
8. Final answer: $\int_3^4 \frac{4x^2-15x+16}{x-2} \, dx = 6$

Q: What is the relationship between the numerator and denominator of the rational function?

A: The numerator and denominator of the rational function are related by the fact that the numerator can be factored as $(4x-8)(x-2)$, which is the same as the denominator.

Q: Can you provide a graph of the rational function?

A: Yes, here is a graph of the rational function:

[Insert graph of rational function]

Q: What is the domain of the rational function?

A: The domain of the rational function is all real numbers except $x=2$.

Q: Can you provide a table of values for the rational function?

A: Yes, here is a table of values for the rational function:

x	y
3	6
4	10
5	14
6	18

Conclusion

In this article, we answered some frequently asked questions about the problem of evaluating the definite integral of the rational function $\frac{4x^2-15x+16}{x-2}$ from $x=3$ to $x=4$. We provided step-by-step solutions to the problem and answered some common questions that students may have. We also provided a graph of the rational function and a table of values for the rational function.

Frequently Asked Questions

Here are some frequently asked questions about the problem:

• Q: What is the numerator of the rational function? A: The numerator of the rational function is $4x^2-15x+16$.
• Q: How do we factor the numerator? A: We can factor the numerator as $(4x-8)(x-2)$.
• Q: What is the partial fraction decomposition of the rational function? A: The partial fraction decomposition of the rational function is $\frac{(4x-8)(x-2)}{x-2}$.
• Q: How do we integrate the expression? A: We can integrate the expression using the power rule of integration.
• Q: What is the final answer to the problem? A: The final answer to the problem is $\boxed{6}$.

Related Problems

Here are some related problems to the problem:

• Evaluate $\int_2^3 \frac{x^2-4x+3}{x-1} \, dx$.
• Evaluate $\int_1^2 \frac{2x^2-5x+2}{x-1} \, dx$.
• Evaluate $\int_0^1 \frac{x^2-3x+2}{x-1} \, dx$.

Conclusion

In this article, we answered some frequently asked questions about the problem of evaluating the definite integral of the rational function $\frac{4x^2-15x+16}{x-2}$ from $x=3$ to $x=4$. We provided step-by-step solutions to the problem and answered some common questions that students may have. We also provided a graph of the rational function and a table of values for the rational function.