When dealing with difficult fractions, partial fraction decomposition is a valuable method for reversing the process of adding fractions with polynomial denominators. This technique can also be applied to integration, making the process more manageable and efficient.

To recap, integration using partial fractions is used for fractions in the form of **(ax + b) / (cx + d)^n**. Before beginning, it is important to determine the degrees of the numerator and denominator. The degree of a polynomial is equal to the highest order term, such as **cx^4** being degree 4 and **dx^8** being degree 8. If the degree of the numerator is less than that of the denominator, we can proceed with partial fraction decomposition.

However, if the degree of the numerator is greater than or equal to the denominator, we must first perform algebraic long division. The next step is to factorize the denominator, starting with linear factors. This involves finding roots using the factor theorem and using these as the linear factors. For denominators with a degree of four or more, quadratic factors must also be considered.

In cases where there are no real roots, such as **x^2 + 4**, we must also check for quadratic factors by multiplying the factors together. For example, **(x^2 + 1) (x^2 + 4) = (x^4 + 5x^2 + 4)**. If the degree of the denominator is six or greater, we may also need to check for cubic factors, but these cases are rare. In general, it is best to focus on denominators with a degree of five or less.

Next, we must consider if there are any repeated factors. This will affect the decomposition of the fraction. If a factor does not repeat, it can be written as a separate numerator. However, if it appears multiple times, all possible multiples must be considered. The degree of the polynomial in the numerator should always be one less than the polynomial in the denominator, except for repeated factors, where it will be the same as the unrepeated root.

Once we have determined the form of the decomposed fraction, we can solve for the unknown coefficients by multiplying through by the denominator and equating equivalent coefficients. For example, to find the partial fraction decomposition of **(x^3 + x^2 + x + 1) / (x^4 + x^2)**, we know the form should be **A / x + B / x^2 + C / (x^2 + 1) + D / (x^4 + x^2)**. By multiplying through by **x^3 (x^2 + 1)**, we get **Ax + Bx^2 + Cx (x^2 + 1) + Dx^3 (x^2 + 1)**, which simplifies to **A + Bx + Cx^3 + Dx^5**. Comparing coefficients, we can solve for **A = 0, B = 1, C = 1**, and **D = 0**. This gives us the partial fraction decomposition of **1 / x^2 + x + 1 / (x^2 + 1)**.

Now, let's see how partial fractions can be applied to integration to simplify the process. For example, to integrate **x / (x^2 - 1)**, first we must perform partial fraction decomposition. By using the difference of two squares, we know that **x^2 - 1 = (x - 1) (x + 1)**. This means the expected form of the partial fraction will be **A / (x - 1) + B / (x + 1)**. By equating the two sides, we get **x = A (x + 1) + B (x - 1)**. Multiplying through by **x^2 - 1**, we get **x = Ax^2 + Ax - Bx^2 + Bx**. This tells us that **A + B = 0** and **A - B = 1**, which leads to **A = 1 / 2**, and **B = -1 / 2**.

Using this information, we can now integrate the fraction easily, resulting in **1 / 2 * ln | x - 1 | - 1 / 2 * ln | x + 1 | + C**. Similarly, to integrate **(x + 2) / (x^2 + 1)**, we can reduce it to **(x + 2) / (x + 2) (x - 2) = 1 / (x - 2)**. Defining **I = 1 / (x - 2)**, and **J = x + 2 / x^2 + 1**, we can use standard integration techniques to solve for **ln | x - 2 |** and **arctan x**. By making a substitution with **u = x^2 + 1**, we can evaluate **K as arctan x + ln (x^2 + 1)**. Combining these, we can integrate to find the solution as **ln | x - 2 | + arctan x + C**.

In summary, integration using partial fractions involves decomposing a fraction and then integrating normally. It should be used when the degree of the polynomial in the numerator is larger than the denominator. By following the steps outlined above, integration using partial fractions can be made more manageable and efficient.

At first, the idea of breaking down a fraction and integrating its individual components may seem daunting and time-consuming. However, incorporating partial fractions into the integration process can greatly improve its overall complexity and manageability.

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