# How do you determine if the improper integral converges or diverges #int_0^oo 1/ (x-2)^2 dx #?

This integral converges to

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The integral is not convergent.

The function:

is not continuous in the interval of integration so we must split the integral as:

Now:

and:

Finally:

Thus the integral is not convergent.

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To determine if the improper integral ( \int_0^\infty \frac{1}{(x-2)^2} , dx ) converges or diverges:

- Evaluate the integral from ( x = 0 ) to ( x = M ), where ( M ) is a large positive number.
- Then, take the limit as ( M ) approaches infinity.

If the limit exists and is a finite number, the integral converges. If the limit does not exist or is infinite, the integral diverges.

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To determine if the improper integral (\int_0^\infty \frac{1}{{(x-2)^2}} , dx) converges or diverges, we need to evaluate its convergence using the properties of improper integrals.

We can break down this improper integral into two parts:

- (\int_0^2 \frac{1}{{(x-2)^2}} , dx)
- (\int_2^\infty \frac{1}{{(x-2)^2}} , dx)

First, let's consider the integral from 0 to 2. At (x = 2), the denominator becomes (0), resulting in an infinite value. Thus, the integral from 0 to 2 diverges.

Next, let's consider the integral from 2 to (\infty). We can evaluate this integral using the limit:

[ \lim_{{a \to \infty}} \int_2^a \frac{1}{{(x-2)^2}} , dx ]

By integrating the function, we get:

[
\lim_{{a \to \infty}} \left[-\frac{1}{{x-2}}\right]*2^a = \lim*{{a \to \infty}} \left(-\frac{1}{{a-2}} + \frac{1}{0}\right)
]

This limit diverges to (\infty).

Since one part of the improper integral diverges and the other part also diverges, the entire improper integral (\int_0^\infty \frac{1}{{(x-2)^2}} , dx) diverges.

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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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