3.5 Q-30

Question Statement

Evaluate the integral:

$$\int \frac{3 x - 8}{(x^2 + x + 2)(x^2 - x + 2)} , dx$$

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Background and Explanation

To solve this integral, we will use the method of partial fraction decomposition. This technique involves expressing the integrand as a sum of simpler fractions that can be easily integrated. In this case, we will decompose the rational function into two simpler fractions, each corresponding to one of the quadratic terms in the denominator.

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Solution

We begin by assuming the following decomposition for the given rational function:

$$\frac{3x - 8}{(x^2 + x + 2)(x^2 - x + 2)} = \frac{A x + B}{x^2 + x + 2} + \frac{C x + D}{x^2 - x + 2}$$

Now, we multiply both sides of this equation by the denominator $(x^2 + x + 2)(x^2 - x + 2)$ to eliminate the fractions:

$$3x - 8 = (A x + B)(x^2 - x + 2) + (C x + D)(x^2 + x + 2)$$

Expanding both sides:

$$3x - 8 = A(x^3 - x^2 + 2x) + B(x^2 - x + 2) + C(x^3 + x^2 + 2x) + D(x^2 + x + 2)$$

Simplifying further:

$$3x - 8 = (A + C) x^3 + (A + B - C + D) x^2 + (2A + B + 2C - D) x + (2B + 2D)$$

Next, we equate the coefficients of like powers of $x$ on both sides:

• For $x^3$: $A + C = 0$
• For $x^2$: $A + B - C + D = 0$
• For $x^1$: $2A + B + 2C - D = 3$
• For the constant term: $2B + 2D = -8$

Now, we solve this system of equations:

1. From $A + C = 0$, we get $C = -A$.
2. Substituting $C = -A$ into the equation $A + B - C + D = 0$ gives:

$$A + B + A + D = 0 \quad \Rightarrow \quad 2A + B + D = 0$$

3. Substituting $C = -A$ into $2A + B + 2C - D = 3$ gives:

$$2A + B - 2A - D = 3 \quad \Rightarrow \quad B - D = 3$$

4. Finally, from $2B + 2D = -8$, we get:

$$B + D = -4$$

Now, solving the system:

• From $B + D = -4$ and $B - D = 3$, adding these two equations gives:

$$2B = -1 \quad \Rightarrow \quad B = -\frac{1}{2}$$

• Substituting $B = -\frac{1}{2}$ into $B + D = -4$:

$$-\frac{1}{2} + D = -4 \quad \Rightarrow \quad D = -\frac{7}{2}$$

• Now, substituting $B = -\frac{1}{2}$ and $D = -\frac{7}{2}$ into $2A + B + D = 0$:

$$2A - \frac{1}{2} - \frac{7}{2} = 0 \quad \Rightarrow \quad 2A - 4 = 0 \quad \Rightarrow \quad A = 2$$

• Since $C = -A$, we find $C = -2$.

Thus, the decomposition becomes:

$$\frac{3x - 8}{(x^2 + x + 2)(x^2 - x + 2)} = \frac{2x - \frac{1}{2}}{x^2 + x + 2} + \frac{-2x - \frac{7}{2}}{x^2 - x + 2}$$

Now, we can integrate term by term:

$$\int \frac{3x - 8}{(x^2 + x + 2)(x^2 - x + 2)} dx = \int \frac{2x - \frac{1}{2}}{x^2 + x + 2} dx + \int \frac{-2x - \frac{7}{2}}{x^2 - x + 2} dx$$

These integrals can be solved using the standard formula for integrals of the form:

$$\int \frac{dx}{x^2 + ax + b} = \frac{1}{\sqrt{b - \frac{a^2}{4}}} \tan^{-1}\left(\frac{2x + a}{\sqrt{b - \frac{a^2}{4}}}\right)$$

After applying this formula, we get:

$$\ln \left| \frac{x^2 - x + 2}{x^2 + x + 2} \right| + \frac{1}{\sqrt{7}} \left[ \tan^{-1}\left( \frac{2x - 1}{\sqrt{7}} \right) + 5 \tan^{-1}\left( \frac{2x + 1}{\sqrt{7}} \right) \right] + C$$

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Key Formulas or Methods Used

1. Partial Fraction Decomposition: This method allows us to express a rational function as a sum of simpler fractions, making it easier to integrate.
2. Integration of Rational Functions: Once the function is decomposed, standard methods of integrating rational functions are applied.
3. Arctangent Formula: Used for integrating terms of the form $\frac{dx}{x^2 + a^2}$.

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Summary of Steps

1. Assume a partial fraction decomposition for the integrand.
2. Multiply both sides by the denominator to clear fractions.
3. Expand the resulting expression and equate coefficients of powers of $x$.
4. Solve the system of equations for the unknowns $A$, $B$, $C$, and $D$.
5. Substitute the values of $A$, $B$, $C$, and $D$ into the decomposed form.
6. Integrate each term using standard integration formulas.
7. Simplify and combine the results to obtain the final answer.