3.5 Q-11

Question Statement

Evaluate the integral:

$$\int \frac{5 x^{2}+9 x+6}{\left(x^{2}-1\right)(2 x+3)} , dx$$

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Background and Explanation

To solve this problem, we will use partial fraction decomposition. This technique allows us to break down a complex rational function into simpler fractions that are easier to integrate.

For the expression $\frac{5 x^{2}+9 x+6}{\left(x^{2}-1\right)(2 x+3)}$, we will express it as a sum of simpler fractions. We need to recognize that $x^2 - 1$ factors as $(x - 1)(x + 1)$.

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Solution

We begin by setting up the partial fraction decomposition:

$$\frac{5 x^{2}+9 x+6}{\left(x^{2}-1\right)(2 x+3)} = \frac{A}{x-1} + \frac{B}{x+1} + \frac{C}{2x+3} \tag{1}$$

Next, multiply both sides of the equation by the denominator $(x-1)(x+1)(2x+3)$:

$$5 x^{2}+9 x+6 = A(x+1)(2x+3) + B(x-1)(2x+3) + C(x-1)(x+1) \tag{2}$$

Step 1: Find $A$

Substitute $x = 1$ into equation (2):

$$5(1)^{2} + 9(1) + 6 = A(1+1)(2(1)+3)$$

Simplify:

$$20 = A(2)(5) \quad \Rightarrow \quad A = \frac{20}{10} = 2$$

Step 2: Find $B$

Substitute $x = -1$ into equation (2):

$$5(-1)^{2} + 9(-1) + 6 = B(-1-1)(2(-1)+3)$$

Simplify:

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

Step 3: Find $C$

Substitute $x = \frac{-3}{2}$ into equation (2):

$$5\left(\frac{-3}{2}\right)^{2} + 9\left(\frac{-3}{2}\right) + 6 = C\left(\frac{-5}{2}\right)\left(\frac{-1}{2}\right)$$

Simplify:

$$\frac{45}{4} - \frac{54}{4} + \frac{24}{4} = \frac{5}{4} C$$ $$\frac{15}{4} = \frac{5}{4} C \quad \Rightarrow \quad C = 3$$

Thus, the partial fraction decomposition is:

$$\frac{5 x^{2}+9 x+6}{\left(x^{2}-1\right)(2 x+3)} = \frac{2}{x-1} - \frac{1}{x+1} + \frac{3}{2x+3}$$

Step 4: Integration

Now, we integrate each term:

$$\int \left( \frac{2}{x-1} - \frac{1}{x+1} + \frac{3}{2x+3} \right) dx$$

Integrating each term:

$$= 2 \ln |x-1| - \ln |x+1| + \frac{3}{2} \ln |2x+3| + C$$

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Key Formulas or Methods Used

• Partial Fraction Decomposition: Decomposing a complex rational function into simpler fractions to facilitate integration.
• Integration of Rational Functions: The integral of $\frac{1}{x}$ is $\ln|x|$, which we used for each term.

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Summary of Steps

1. Set up the partial fraction decomposition: $\frac{5 x^{2}+9 x+6}{(x^2 - 1)(2x+3)} = \frac{A}{x-1} + \frac{B}{x+1} + \frac{C}{2x+3}$

2. Multiply both sides by the denominator to obtain an equation.

3. Solve for $A$, $B$, and $C$ by substituting suitable values of $x$.

4. Perform the integration on each term separately.