# How do you integrate #x^2/sqrt(9-x^2) dx#?

You can do this with trig substitution. Notice how this is of the form

which looks like

while

So, let:

That gives

Now we just have

Use the identity

to then get

Thus, our final result is

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To integrate ( \frac{x^2}{\sqrt{9-x^2}} ) with respect to ( x ), you can use trigonometric substitution. Let ( x = 3\sin(\theta) ), ( dx = 3\cos(\theta) , d\theta ). After substitution and simplification, the integral becomes ( \int \frac{9\sin^2(\theta)}{3\cos(\theta)} \cdot 3\cos(\theta) , d\theta ). This simplifies to ( \int 3\sin^2(\theta) , d\theta ). Now, use the identity ( \sin^2(\theta) = \frac{1 - \cos(2\theta)}{2} ) to rewrite the integral, giving ( \frac{1}{2} \int (3 - 3\cos(2\theta)) , d\theta ). Integrate each term separately to obtain ( \frac{1}{2} (3\theta - \frac{3}{2}\sin(2\theta)) + C ). Finally, substitute ( \theta = \sin^{-1}(\frac{x}{3}) ) and simplify to get the final answer.

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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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