∫ dx/(x²-a²)

Given:

The integral to solve is: ∫ dx / (x² - a²)

Concept Used:

The integral of the given type can be solved using partial fraction decomposition and standard integral formulas. The standard integral is:

∫ dx / (x² - a²) = (1 / 2a) log|(x - a) / (x + a)| + C

Here:

• x: Variable of integration

• a: A constant

• log: Refers to the natural logarithm (ln)

Calculation:

We rewrite the given integral using partial fraction decomposition:

1 / (x² - a²) = 1 / [(x - a)(x + a)]

Using partial fractions:

1 / [(x - a)(x + a)] = A / (x - a) + B / (x + a)

Solving for A and B, we find:

A = 1 / 2a, B = -1 / 2a

Substituting back:

1 / [(x - a)(x + a)] = (1 / 2a) / (x - a) - (1 / 2a) / (x + a)

Integrating term by term:

∫ dx / (x² - a²) = (1 / 2a) ∫ dx / (x - a) - (1 / 2a) ∫ dx / (x + a)

⇒ (1 / 2a) log|x - a| - (1 / 2a) log|x + a| + C

⇒ (1 / 2a) log|(x - a) / (x + a)| + C