Evaluate The Integral:$\[ \int \frac{3x^2 + 4x + 1}{3x^3 + 6x^2 + 3x + 5} \, Dx \\]

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Introduction

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In this article, we will delve into the world of calculus and evaluate the given integral. The integral in question is ${ \int \frac{3x^2 + 4x + 1}{3x^3 + 6x^2 + 3x + 5} , dx }$. This type of integral is known as a rational function, and it can be challenging to evaluate. However, with the right approach and techniques, we can break it down and find the solution.

Understanding the Integral

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Before we dive into the solution, let's take a closer look at the integral. The numerator is a quadratic function, $3x^2 + 4x + 1$, and the denominator is a cubic function, $3x^3 + 6x^2 + 3x + 5$. Our goal is to find the antiderivative of this rational function.

Step 1: Factor the Denominator

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To evaluate this integral, we need to factor the denominator. The denominator can be factored as follows:

$$3x^3 + 6x^2 + 3x + 5 = (3x + 1)(x^2 + 2x + 5)$$

Step 2: Rewrite the Integral

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Now that we have factored the denominator, we can rewrite the integral as follows:

$$\int \frac{3x^2 + 4x + 1}{(3x + 1)(x^2 + 2x + 5)} \, dx$$

Step 3: Use Partial Fraction Decomposition

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To evaluate this integral, we can use partial fraction decomposition. We can write the rational function as the sum of two simpler fractions:

$$\frac{3x^2 + 4x + 1}{(3x + 1)(x^2 + 2x + 5)} = \frac{A}{3x + 1} + \frac{Bx + C}{x^2 + 2x + 5}$$

Step 4: Find the Values of A, B, and C

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To find the values of A, B, and C, we can multiply both sides of the equation by the denominator and then equate the coefficients of the numerator.

$$3x^2 + 4x + 1 = A(x^2 + 2x + 5) + (3x + 1)(Bx + C)$$

Step 5: Solve for A, B, and C

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By equating the coefficients of the numerator, we can solve for A, B, and C.

$$A = \frac{3}{3} = 1$$

$$B = \frac{4 - 2}{3} = \frac{2}{3}$$

$$C = \frac{1 - 5}{3} = -\frac{4}{3}$$

Step 6: Rewrite the Integral

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Now that we have found the values of A, B, and C, we can rewrite the integral as follows:

$$\int \frac{3x^2 + 4x + 1}{(3x + 1)(x^2 + 2x + 5)} \, dx = \int \frac{1}{3x + 1} \, dx + \int \frac{\frac{2}{3}x - \frac{4}{3}}{x^2 + 2x + 5} \, dx$$

Step 7: Evaluate the Integrals

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Now that we have rewritten the integral, we can evaluate the two separate integrals.

$$\int \frac{1}{3x + 1} \, dx = \frac{1}{3} \ln |3x + 1| + C_1$$

$$\int \frac{\frac{2}{3}x - \frac{4}{3}}{x^2 + 2x + 5} \, dx = \frac{1}{3} \ln |x^2 + 2x + 5| - \frac{2}{3} \arctan \left( \frac{2x + 1}{3} \right) + C_2$$

Step 8: Combine the Results

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Now that we have evaluated the two separate integrals, we can combine the results to find the final answer.

$$\int \frac{3x^2 + 4x + 1}{3x^3 + 6x^2 + 3x + 5} \, dx = \frac{1}{3} \ln |3x + 1| + \frac{1}{3} \ln |x^2 + 2x + 5| - \frac{2}{3} \arctan \left( \frac{2x + 1}{3} \right) + C$$

Conclusion

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In this article, we have evaluated the given integral using partial fraction decomposition. We have broken down the integral into two simpler fractions and then evaluated each fraction separately. The final answer is a combination of the two separate integrals.

Final Answer

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The final answer is:

$$\boxed{\frac{1}{3} \ln |3x + 1| + \frac{1}{3} \ln |x^2 + 2x + 5| - \frac{2}{3} \arctan \left( \frac{2x + 1}{3} \right) + C}$$

References

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• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart
• [3] Partial Fractions, Wolfram MathWorld

Tags

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• calculus
• integral
• partial fraction decomposition
• rational function
• antiderivative
• logarithmic function
• arctangent function
  

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Introduction

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In our previous article, we evaluated the integral ${ \int \frac{3x^2 + 4x + 1}{3x^3 + 6x^2 + 3x + 5} , dx }$. In this article, we will answer some frequently asked questions related to this integral.

Q: What is the purpose of partial fraction decomposition?

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A: Partial fraction decomposition is a technique used to break down a rational function into simpler fractions. This technique is useful when evaluating integrals of rational functions.

Q: How do I determine the values of A, B, and C in partial fraction decomposition?

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A: To determine the values of A, B, and C, you need to multiply both sides of the equation by the denominator and then equate the coefficients of the numerator.

Q: What is the difference between a rational function and a polynomial?

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A: A rational function is a function that can be expressed as the ratio of two polynomials. A polynomial is a function that can be expressed as a sum of terms, where each term is a constant or a variable raised to a non-negative integer power.

Q: Can I use partial fraction decomposition to evaluate any integral?

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A: No, partial fraction decomposition can only be used to evaluate integrals of rational functions. If the integral is not a rational function, you may need to use other techniques, such as substitution or integration by parts.

Q: How do I know if an integral is a rational function?

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A: An integral is a rational function if it can be expressed as the ratio of two polynomials. You can check if an integral is a rational function by looking at the numerator and denominator. If both the numerator and denominator are polynomials, then the integral is a rational function.

Q: Can I use partial fraction decomposition to evaluate integrals with complex numbers?

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A: Yes, partial fraction decomposition can be used to evaluate integrals with complex numbers. However, you need to be careful when working with complex numbers, as they can be tricky to handle.

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

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A: The constant C is a constant of integration, which means that it can take on any value. The constant C is used to represent the fact that the integral is not unique, and that there are many possible antiderivatives.

Q: Can I use partial fraction decomposition to evaluate integrals with trigonometric functions?

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A: No, partial fraction decomposition is not typically used to evaluate integrals with trigonometric functions. However, you can use other techniques, such as substitution or integration by parts, to evaluate integrals with trigonometric functions.

Q: How do I know if an integral is a rational function with trigonometric functions?

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A: An integral is a rational function with trigonometric functions if it can be expressed as the ratio of two polynomials, where one or both of the polynomials contain trigonometric functions. You can check if an integral is a rational function with trigonometric functions by looking at the numerator and denominator.

Q: Can I use partial fraction decomposition to evaluate integrals with exponential functions?

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A: No, partial fraction decomposition is not typically used to evaluate integrals with exponential functions. However, you can use other techniques, such as substitution or integration by parts, to evaluate integrals with exponential functions.

Q: How do I know if an integral is a rational function with exponential functions?

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A: An integral is a rational function with exponential functions if it can be expressed as the ratio of two polynomials, where one or both of the polynomials contain exponential functions. You can check if an integral is a rational function with exponential functions by looking at the numerator and denominator.

Conclusion

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In this article, we have answered some frequently asked questions related to evaluating the integral ${ \int \frac{3x^2 + 4x + 1}{3x^3 + 6x^2 + 3x + 5} , dx }$. We hope that this article has been helpful in clarifying any doubts you may have had about this integral.

Final Answer

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The final answer is:

$$\boxed{\frac{1}{3} \ln |3x + 1| + \frac{1}{3} \ln |x^2 + 2x + 5| - \frac{2}{3} \arctan \left( \frac{2x + 1}{3} \right) + C}$$

References

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• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart
• [3] Partial Fractions, Wolfram MathWorld

Tags

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• calculus
• integral
• partial fraction decomposition
• rational function
• antiderivative
• logarithmic function
• arctangent function
• trigonometric function
• exponential function