3.5 Q-13

Question Statement

We are asked to solve the following integral:

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

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Background and Explanation

This problem involves partial fraction decomposition, a method used to simplify a rational function into a sum of simpler fractions. Each fraction is easier to integrate individually. In this case, the denominator has a repeated factor, $(x - 1)^2$, and a linear factor, $(x + 1)$, so we will decompose the expression into the form:

$$\frac{A}{x + 1} + \frac{B}{x - 1} + \frac{C}{(x - 1)^2}$$

We then solve for the constants $A$, $B$, and $C$ by equating terms.

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Solution

We begin by expressing the integrand as a sum of partial fractions:

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

Now, multiply both sides of equation (i) by $(x - 1)^2(x + 1)$ to eliminate the denominators:

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

Step 1: Solve for $B$

Substitute $x = 1$ into equation (ii) to solve for $B$:

$$2(1)^2 = B(1 + 1) 2 = B(2) \quad \Rightarrow \quad B = 1$$

Step 2: Solve for $C$

Substitute $x = -1$ into equation (ii) to solve for $C$:

$$2(-1)^2 = C(-1 - 1)^2 2 = C(-2)^2 \quad \Rightarrow \quad 2 = 4C C = \frac{1}{2}$$

Step 3: Solve for $A$

Now, equate the coefficients of $x^2$ terms from both sides of equation (ii) to solve for $A$:

$$2 = A + C 2 = A + \frac{1}{2} A = \frac{3}{2}$$

Thus, we can rewrite the original integral as:

$$\int \frac{2x^2}{(x - 1)^2(x + 1)} , dx = \int \left[ \frac{\frac{3}{2}}{x - 1} + \frac{1}{(x - 1)^2} + \frac{\frac{1}{2}}{x + 1} \right] , dx$$

Step 4: Integrate the terms

Now we integrate each term:

1. $\int \frac{\frac{3}{2}}{x - 1} , dx = \frac{3}{2} \ln |x - 1|$
2. $\int \frac{1}{(x - 1)^2} , dx = -\frac{1}{x - 1}$
3. $\int \frac{\frac{1}{2}}{x + 1} , dx = \frac{1}{2} \ln |x + 1|$

Thus, the integral is:

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

Final Answer:

Therefore, the solution to the integral is:

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

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Key Formulas or Methods Used

• Partial Fraction Decomposition: Used to break the rational function into simpler fractions.
• Integration of Common Rational Functions:
  • $\int \frac{1}{x} , dx = \ln |x|$
  • $\int \frac{1}{x^2} , dx = -\frac{1}{x}$

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Summary of Steps

1. Express the integrand as a sum of partial fractions.
2. Solve for the unknown coefficients $A$, $B$, and $C$ by substituting values for $x$.
3. Rewrite the integral with the solved coefficients.
4. Integrate each term using standard integration rules.
5. Combine the results and simplify the expression.