Integration of 1 / (x² - a²)

The exact mathematical formula is:

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

(Where 'a' is a massive constant, 'log' is the natural logarithm base 'e', and 'C' is the mandatory constant of integration).

To violently prove this formula, we must break the heavy denominator into massive pieces using the algebraic identity: x² - a² = (x - a)(x + a).

Step 1: Write the expression as: 1 / [(x - a)(x + a)]

Step 2: Use heavy Partial Fractions: 1 / [(x - a)(x + a)] = A / (x - a) + B / (x + a) By solving mathematically, we heavily find that A = 1/2a and B = -1/2a.

Step 3: Substitute back into the massive integral: ∫ [ (1/2a) / (x - a) - (1/2a) / (x + a) ] dx

Step 4: Pull out the massive constant (1/2a): (1/2a) * [ ∫ 1/(x - a) dx - ∫ 1/(x + a) dx ]

Step 5: Apply the massive log formula (∫ 1/x = log|x|): (1/2a) * [ log|x - a| - log|x + a| ]

Step 6: Use the massive log property (log m - log n = log(m/n)): = (1 / 2a) * log |(x - a) / (x + a)| + C. (Hence Proved).

Evaluate: ∫ 1 / (x² - 16) dx

• Here, we can write 16 as 4². So, our massive constant a = 4.
• Directly heavily apply the formula:
• = (1 / 2(4)) * log |(x - 4) / (x + 4)| + C
• = (1/8) * log |(x - 4) / (x + 4)| + C