3.5 Q-25

Question Statement

Evaluate the following integral:

$$\int \frac{2 x^{2}-x-7}{(x+2)^{2}(x^{2}+x+1)} , dx$$

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Background and Explanation

This problem involves integrating a rational function, which can be simplified using partial fraction decomposition. The method of partial fractions helps express a complex fraction as the sum of simpler fractions, making integration easier. In this case, we decompose the given rational function into terms that can be integrated directly.

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Solution

We begin by setting up the partial fraction decomposition:

$$\frac{2 x^{2} - x - 7}{(x+2)^{2}(x^{2}+x+1)} = \frac{A}{x+2} + \frac{B}{(x+2)^{2}} + \frac{C x + D}{x^{2}+x+1} \tag{i}$$

Step 1: Multiply both sides by $(x^2+x+1)(x+2)^2$ to clear the denominators:

$$2x^2 - x - 7 = A(x+2)(x^2 + x + 1) + B(x^2 + x + 1) + (Cx + D)(x+2)^2 \tag{ii}$$

Step 2: Set $x = -2$ to solve for $B$:

Substituting $x = -2$ into equation (ii):

$$2(-2)^2 - (-2) - 7 = B((-2)^2 - 2 + 1)$$

Simplifying:

$$8 + 2 - 7 = B(4 - 2 + 1) \quad \Rightarrow \quad B = 1$$

Step 3: Compare coefficients of powers of $x$ on both sides to solve for $A$, $C$, and $D$:

From equation (ii), we compare the coefficients of $x^3$, $x^2$, and $x$:

1. $0 = A + C$ $\Rightarrow C = -A$
2. $2 = 3A + B + 4C + D$ $\Rightarrow 2 = 3A + 1 + 4(-A) + D$
3. $-1 = 3A + B + 4C + 4D$
4. $-7 = 2A + B + 4D$

By solving these equations, we find:

• $A = -2$
• $C = 2$
• $D = -1$

Step 4: Substitute the values of $A$, $B$, $C$, and $D$ into the original decomposition:

$$\int \frac{2 x^{2} - x - 7}{(x+2)^{2}(x^{2}+x+1)} , dx = \int \left( \frac{-2}{x+2} + \frac{1}{(x+2)^2} + \frac{2x-1}{x^2+x+1} \right) dx$$

Step 5: Integrate each term:

• The integral of $\frac{-2}{x+2}$ is $-2 \ln |x+2|$
• The integral of $\frac{1}{(x+2)^2}$ is $-\frac{1}{x+2}$
• The integral of $\frac{2x+1}{x^2+x+1}$ is $\ln(x^2 + x + 1)$
• The integral of $\frac{2}{\left(x + \frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2}$ is $\frac{4}{\sqrt{3}} \tan^{-1} \left( \frac{x + \frac{1}{2}}{\frac{\sqrt{3}}{2}} \right)$

Final Answer:

Thus, the integral evaluates to:

$$\int \frac{2 x^{2} - x - 7}{(x+2)^{2}(x^{2}+x+1)} , dx = 2 \ln |x+2| - \frac{1}{x+2} + \ln(x^2 + x + 1) - \frac{4}{\sqrt{3}} \tan^{-1} \left( \frac{2x+1}{\sqrt{3}} \right) + C$$

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Key Formulas or Methods Used

• Partial Fraction Decomposition: Used to express the rational function as the sum of simpler fractions.
• Basic Integration Rules: For rational functions, the integral of $\frac{1}{x+a}$ is $\ln |x+a|$, and the integral of $\frac{1}{(x+a)^2}$ is $\frac{-1}{x+a}$.
• Arctangent Integral: The integral of $\frac{1}{(x+h)^2 + a^2}$ is $\frac{1}{a} \tan^{-1} \left( \frac{x+h}{a} \right)$.

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Summary of Steps

1. Set up the partial fraction decomposition of the rational function.
2. Multiply both sides by the common denominator to eliminate fractions.
3. Substitute $x = -2$ to find $B$.
4. Compare coefficients of powers of $x$ to solve for $A$, $C$, and $D$.
5. Substitute values of $A$, $B$, $C$, and $D$ into the partial fraction decomposition.
6. Integrate each term and combine the results.
7. Write the final solution, including the constant of integration $C$.