Integration of 1 / (x² - a²)

In massive Class 12 Calculus, evaluating the integral of rational functions is a highly critical skill. One of the most famous, massive standard formulas you must physically memorize for the board exams is the integration of the function 1 / (x² - a²).

Question (Click to Flip)

Can I solve this using trigonometric substitution?

Answer

Yes! Instead of massive partial fractions, you can substitute x = a * sec(θ). However, the partial fraction method is massively faster and far less confusing for board exams.

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Key Facts

If the massive formula is reversed to 1 / (a² - x²), the formula slightly changes! It becomes: (1/2a) * log |(a + x) / (a - x)| + C.

You absolutely must include the massive Modulus bars '| |' around the log argument, because the natural logarithm of a mathematically negative number is completely physically undefined.

1. The Standard Massive Formula

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).

2. The Step-by-Step Proof (Using Partial Fractions)

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).

3. A Quick Example

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

Questions and Answers

Can I solve this using trigonometric substitution?+

Yes! Instead of massive partial fractions, you can substitute **x = a * sec(θ)**. However, the partial fraction method is massively faster and far less confusing for board exams.

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