# How do you evaluate the integral #int lnabs(x-5)dx# from 0 to 2 if it converges?

Evaluate the indefinite integral by parts, considering that:

so:

Now we have:

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To evaluate the integral ∫ ln|𝑥−5| 𝑑𝑥 from 0 to 2, first split the integral into two parts: from 0 to 5 and from 5 to 2. Then, integrate each part separately using the property ln|𝑎| = ln(𝑎) if 𝑎 > 0 and ln|𝑎| = ln(−𝑎) if 𝑎 < 0. Evaluate the antiderivative of ln|𝑥−5| over each interval and subtract the result of the lower limit from the upper limit.

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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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