# Integration of 1/(x^2-a^2) Using Substitution Method

• qw3x
In summary, the conversation discusses finding the integral of \frac{dx}{x^2-a^2} using substitution. The poster initially solved it using partial fractions, but their professor wants them to use substitution. The poster suggests using x=asec(\theta) as the substitution, and another user confirms that it should work.
qw3x

## Homework Statement

$\int \frac{dx}{x^2-a^2}$

## The Attempt at a Solution

I've reached the answer, $\frac{1}{2a} ln |\frac{x-a}{x+a}| + C$ , using partial fractions, but my professor asks for the work using substitution. Now I know how to do this when there's a radical in the denominator, but would this also be a substitution $x=asec(\theta)$?

Try $x=a\cosh u$

qw3x said:

## Homework Statement

$\int \frac{dx}{x^2-a^2}$

## The Attempt at a Solution

I've reached the answer, $\frac{1}{2a} ln |\frac{x-a}{x+a}| + C$ , using partial fractions, but my professor asks for the work using substitution. Now I know how to do this when there's a radical in the denominator, but would this also be a substitution $x=asec(\theta)$?
Hello qw3x. Welcome to PF !

$x=asec(\theta)$ should work fine. I think that leads to integrating csc(θ) .

## 1. What is the general formula for integrating 1/(x^2-a^2)?

The general formula for integrating 1/(x^2-a^2) is ln|x+a|/2a + ln|x-a|/(-2a) + C, where C is the constant of integration.

## 2. How do I solve for the integral of 1/(x^2-a^2) if a is equal to 0?

If a is equal to 0, the integral becomes ln|x| + C. This is because the term 1/(x^2-a^2) simplifies to 1/x^2 when a is equal to 0.

## 3. Can I use substitution to solve for the integral of 1/(x^2-a^2)?

Yes, substitution can be used to solve for the integral. A common substitution is u = x + a, which simplifies the integral to ln|u|/2a + C.

## 4. Is there a specific method for integrating 1/(x^2-a^2) if a is a complex number?

If a is a complex number, the integral can be solved using partial fractions and the inverse tangent function. The final answer will include both real and imaginary components.

## 5. Can the integral of 1/(x^2-a^2) be solved using any other methods?

Yes, the integral can also be solved using trigonometric substitution, where the substitution x = a tan(u) is made. This method is useful when a^2 is a perfect square.

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