# How do you evaluate the definite integral #int (3/x^2-1)dx# from [1,2]?

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To evaluate the definite integral ∫(3/(x^2 - 1))dx from 1 to 2, you can use the method of partial fraction decomposition to rewrite the integrand as a sum of simpler fractions.

The integrand can be expressed as 3/(x^2 - 1) = 3/((x - 1)(x + 1)).

Using partial fraction decomposition, we can write 3/((x - 1)(x + 1)) = A/(x - 1) + B/(x + 1), where A and B are constants.

By finding common denominators and equating coefficients, we find that A = 1 and B = -1.

So, the integral becomes ∫(1/(x - 1) - 1/(x + 1))dx from 1 to 2.

Integrating each term separately, we get ∫(1/(x - 1) - 1/(x + 1))dx = ln|x - 1| - ln|x + 1| evaluated from 1 to 2.

Substituting the limits of integration, we have ln|2 - 1| - ln|2 + 1| - (ln|1 - 1| - ln|1 + 1|).

Simplifying further, we get ln(1) - ln(3) - (ln(0) - ln(2)).

Since ln(0) is undefined, this term contributes -∞ to the integral.

Thus, the integral is evaluated as ln(1) - ln(3) - (-∞).

ln(1) is 0, and ln(3) is a finite number.

So, the integral evaluates to -∞ - 0 + ln(3).

Therefore, the definite integral ∫(3/(x^2 - 1))dx from 1 to 2 is -∞ + ln(3).

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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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