# How do you find the integral of #int x / (x^2-9) dx# from 1 to infinity?

That integral diverges.

The first of the integrals is:

The integral diverges.

Because one of the integrals diverges, the entire integral diverges.

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To find the integral of ( \int \frac{x}{x^2 - 9} , dx ) from 1 to infinity, we first perform partial fraction decomposition on the integrand:

[ \frac{x}{x^2 - 9} = \frac{x}{(x + 3)(x - 3)} = \frac{A}{x + 3} + \frac{B}{x - 3} ]

Solving for ( A ) and ( B ), we find:

[ A = \frac{1}{6} ] [ B = -\frac{1}{6} ]

So, the integral becomes:

[ \int \frac{x}{x^2 - 9} , dx = \int \left( \frac{1}{6} \cdot \frac{1}{x + 3} - \frac{1}{6} \cdot \frac{1}{x - 3} \right) , dx ]

Integrating term by term, we get:

[ \int \frac{x}{x^2 - 9} , dx = \frac{1}{6} \ln|x + 3| - \frac{1}{6} \ln|x - 3| + C ]

Now, to find the integral from 1 to infinity, we take the limit as ( b ) approaches infinity of the definite integral from 1 to ( b ):

[ \lim_{{b \to \infty}} \left( \frac{1}{6} \ln|b + 3| - \frac{1}{6} \ln|b - 3| \right) - \left( \frac{1}{6} \ln|1 + 3| - \frac{1}{6} \ln|1 - 3| \right) ]

Simplifying, we get:

[ \lim_{{b \to \infty}} \left( \frac{1}{6} \ln|b + 3| - \frac{1}{6} \ln|b - 3| - \frac{1}{6} \ln 4 + \frac{1}{6} \ln 2 \right) ]

Since ( \ln|b + 3| ) and ( \ln|b - 3| ) approach infinity as ( b ) approaches infinity, the limit becomes:

[ \lim_{{b \to \infty}} \left( \frac{1}{6} \ln b - \frac{1}{6} \ln b \right) = 0 ]

Therefore, the integral from 1 to infinity of ( \frac{x}{x^2 - 9} , dx ) is 0.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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