# How do you integrate #1/(x^2 - 3)#?

Thus:

Thus:

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This is a common integral:

This is a proper final answer. However, we can simplify it rather sneakily:

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To integrate ( \frac{1}{{x^2 - 3}} ), you can use a technique called partial fraction decomposition. First, factor the denominator:

[ x^2 - 3 = (x - \sqrt{3})(x + \sqrt{3}) ]

Then, express ( \frac{1}{{x^2 - 3}} ) as the sum of two fractions:

[ \frac{1}{{x^2 - 3}} = \frac{A}{{x - \sqrt{3}}} + \frac{B}{{x + \sqrt{3}}} ]

Find the values of ( A ) and ( B ) by equating numerators:

[ 1 = A(x + \sqrt{3}) + B(x - \sqrt{3}) ]

Solve for ( A ) and ( B ), which yields ( A = \frac{1}{2\sqrt{3}} ) and ( B = -\frac{1}{2\sqrt{3}} ).

Now, integrate each fraction separately:

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

Where ( C ) is the constant of integration.

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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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