# How do you find the indefinite integral of #int x^2/(3-x^2)#?

For the blue bit, we will use the hyperbolic identity:

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To find the indefinite integral of ( \int \frac{x^2}{3-x^2} ), you can use partial fraction decomposition. The integral becomes:

[ \int \frac{x^2}{3-x^2} , dx = \int \frac{x^2}{(x-\sqrt{3})(x+\sqrt{3})} , dx ]

Now, perform partial fraction decomposition:

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

Solving for ( A ) and ( B ), you'll find:

[ A = -\frac{\sqrt{3}}{6}, \quad B = \frac{\sqrt{3}}{6} ]

Now, integrate each term separately:

[ \int \frac{x^2}{3-x^2} , dx = -\frac{\sqrt{3}}{6} \ln|x-\sqrt{3}| + \frac{\sqrt{3}}{6} \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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