3.5 Q-6

Question Statement

Evaluate the integral:
$\int \frac{2x}{x^2 - a^2} , dx$

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Background and Explanation

The given integral involves a rational function with a quadratic denominator that factors into two linear terms. The method of partial fraction decomposition is used to break down the fraction into simpler components. These components can then be integrated individually using logarithmic integration.

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Solution

Step 1: Factorize the Denominator

The denominator $x^2 - a^2$ is a difference of squares, which can be factored as: $(x^2 - a^2) = (x - a)(x + a)$

Thus, the integral becomes: $\int \frac{2x}{x^2 - a^2} , dx = \int \frac{2x}{(x - a)(x + a)} , dx$

Step 2: Partial Fraction Decomposition

Express $\frac{2x}{(x - a)(x + a)}$ as: $\frac{2x}{(x - a)(x + a)} = \frac{A}{x - a} + \frac{B}{x + a}$

Multiply through by $(x - a)(x + a)$ to clear the denominators: $2x = A(x + a) + B(x - a)$

Expand and group terms: $2x = Ax + Aa + Bx - Ba$ $2x = (A + B)x + (Aa - Ba)$

Equating coefficients of like terms:

1. Coefficient of $x$: $A + B = 2$
2. Constant term: $Aa - Ba = 0 \quad \Rightarrow \quad a(A - B) = 0$

Since $a \neq 0$, it follows that: $A - B = 0 \quad \Rightarrow \quad A = B$

Substitute $A = B$ into $A + B = 2$: $A + A = 2 \quad \Rightarrow \quad A = B = 1$

Step 3: Rewrite the Integral

Substituting $A = 1$ and $B = 1$: $\frac{2x}{(x - a)(x + a)} = \frac{1}{x - a} + \frac{1}{x + a}$

Thus, the integral becomes: $\int \frac{2x}{x^2 - a^2} , dx = \int \frac{1}{x - a} , dx + \int \frac{1}{x + a} , dx$

Step 4: Integrate Each Term

1. For the first term: $\int \frac{1}{x - a} , dx = \ln|x - a|$

2. For the second term: $\int \frac{1}{x + a} , dx = \ln|x + a|$

Adding these results: $\int \frac{2x}{x^2 - a^2} , dx = \ln|x - a| + \ln|x + a| + C$

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Key Formulas or Methods Used

1. Partial Fraction Decomposition:
   $\frac{2x}{(x - a)(x + a)} = \frac{A}{x - a} + \frac{B}{x + a}$

2. Logarithmic Integration:
   $\int \frac{1}{x - a} , dx = \ln|x - a|, \quad \int \frac{1}{x + a} , dx = \ln|x + a|$

3. Difference of Squares Factorization:
   $x^2 - a^2 = (x - a)(x + a)$

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Summary of Steps

1. Factorize the denominator:
   $x^2 - a^2 = (x - a)(x + a)$
2. Apply partial fraction decomposition:
   $\frac{2x}{x^2 - a^2} = \frac{1}{x - a} + \frac{1}{x + a}$
3. Rewrite the integral using the decomposed form.
4. Integrate each term:
   $\ln|x - a| \quad \text{and} \quad \ln|x + a|$
5. Combine results and add the constant of integration.

Final Answer:

$\boxed{\ln|x - a| + \ln|x + a| + C}$