# How do you integrate #(1-x^2)^.5#?

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To integrate ( \sqrt{1 - x^2} ), you can use the trigonometric substitution method.

Let ( x = \sin(\theta) ), then ( dx = \cos(\theta) , d\theta ).

Substitute ( x = \sin(\theta) ) into ( \sqrt{1 - x^2} ):

[ \sqrt{1 - x^2} = \sqrt{1 - \sin^2(\theta)} = \sqrt{\cos^2(\theta)} = \cos(\theta) ]

Now, integrate ( \cos(\theta) , d\theta ):

[ \int \cos(\theta) , d\theta = \sin(\theta) + C ]

Substitute back ( x = \sin(\theta) ) to get the final result:

[ \int \sqrt{1 - x^2} , dx = \sin(\theta) + C ]

Remembering that ( x = \sin(\theta) ), you can write:

[ \int \sqrt{1 - x^2} , dx = \sin^{-1}(x) + C ]

So, the integral of ( \sqrt{1 - x^2} ) is ( \sin^{-1}(x) + 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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