Hello qw3x. Welcome to PF !qw3x said:Homework Statement
[itex]\int \frac{dx}{x^2-a^2}[/itex]
Homework Equations
The Attempt at a Solution
I've reached the answer, [itex]\frac{1}{2a} ln |\frac{x-a}{x+a}| + C[/itex] , 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 [itex]x=asec(\theta)[/itex]?
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.
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.
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.
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.
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.