Calculation Integral of 3x + 2 / x(x + 2)^2 + 16x dx

 Introduction

In the world of mathematics, integration is a powerful tool used to find areas, volumes, and accumulated quantities in a wide range of applications, from physics to engineering to economics. Solving integrals often involves techniques like substitution and partial fraction decomposition, which allow us to simplify complex expressions and find their antiderivatives. In this article, we will explore the step-by-step process of calculating the integral of $\frac{3�+2}{�(�+2{)}^2+16�}$ using partial fraction decomposition.

Part 1: The Integral Challenge

The integral we are tasked with is:

$$\int \frac{3�+2}{�(�+2{)}^2+16�}��$$

At first glance, this expression may seem intimidating. However, by employing a strategic approach, we can break it down into more manageable components.

Part 2: Partial Fraction Decomposition

To simplify the given integral, we'll start by performing partial fraction decomposition. This technique involves expressing a complex fraction as a sum of simpler fractions with unknown constants. In this case, we want to break down the expression into the following form:

$$\frac{3�+2}{�(�+2)(�+8)}=\frac{�}{�}+\frac{�}{�+2}+\frac{�}{�+8}$$

Here, $�$, $�$, and $�$ are constants that we need to determine. To find these constants, we'll clear the denominators by multiplying both sides of the equation by $�(�+2)(�+8)$:

$$3�+2=�(�+2)(�+8)+��(�+8)+��(�+2)$$

Now, to find the values of $�$, $�$, and $�$, we'll choose strategic values for $�$ that simplify the equation. Let's consider three values: $�=0$, $�=-2$, and $�=-8$.

1. When $�=0$:

$$2=16�⟹�=\frac{1}{8}$$

2. When $�=-2$:

$$-4=-6�⟹�=\frac{2}{3}$$

3. When $�=-8$:

$$-22=-60�⟹�=\frac{11}{30}$$

Now that we have found the values of $�$, $�$, and $�$, we can express the integral as:

$$\int (\frac{1}{8�}+\frac{2}{3(�+2)}+\frac{11}{30(�+8)})��$$

With the expression in this form, we can proceed to integrate each term separately.

Part 3: Integrating Each Term

1. Integral of $\frac{1}{8�}$ with respect to $�$:

$$\int \frac{1}{8�}��=\frac{1}{8}\ln \mathrm{\mid }�\mathrm{\mid }+{�}_1$$

Here, ${�}_1$ represents the constant of integration.

2. Integral of $\frac{2}{3(�+2)}$ with respect to $�$:

$$\int \frac{2}{3(�+2)}��=\frac{2}{3}\ln \mathrm{\mid }�+2\mathrm{\mid }+{�}_2$$

In this case, ${�}_2$ is the constant of integration.

3. Integral of $\frac{11}{30(�+8)}$ with respect to $�$:

$$\int \frac{11}{30(�+8)}��=\frac{11}{30}\ln \mathrm{\mid }�+8\mathrm{\mid }+{�}_3$$

Similarly, ${�}_3$ is the constant of integration.

Part 4: Combining the Results

Now that we have integrated each term separately, we can combine the results to express the final integral:

$$\int (\frac{1}{8�}+\frac{2}{3(�+2)}+\frac{11}{30(�+8)})��=\frac{1}{8}\ln \mathrm{\mid }�\mathrm{\mid }+\frac{2}{3}\ln \mathrm{\mid }�+2\mathrm{\mid }+\frac{11}{30}\ln \mathrm{\mid }�+8\mathrm{\mid }+�$$

Here, $�$ represents the constant of integration, which is the sum of ${�}_1$, ${�}_2$, and ${�}_3$.

Conclusion

In this article, we embarked on a mathematical journey to calculate the integral of $\frac{3�+2}{�(�+2{)}^2+16�}$ by employing partial fraction decomposition. By factoring the denominator and expressing the complex fraction as a sum of simpler fractions, we were able to determine the values of the constants $�$, $�$, and $�$. Subsequently, we integrated each term separately and combined the results to obtain the final expression for the integral.

Integration, as demonstrated in this example, is a fundamental mathematical tool with diverse applications in various fields, including physics, engineering, economics, and more. Understanding integration techniques allows us to tackle complex problems and gain valuable insights into real-world phenomena.