# How do you integrate #(x^5)(sqrt(4 - x^2)) dx#?

This is of the form:

which looks like:

So, let's do the following substitution. Let:

thus:

We can then get:

Thus, we can re-substitute back in the previous values.

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Use a

Substituting in the integral yields:

Now simplify and rewrite as you see fit.

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To integrate ((x^5)(\sqrt{4 - x^2}) , dx), you can use the substitution method. Let (u = 4 - x^2), then (du = -2x , dx). Solving for (x , dx), you get (x , dx = -\frac{1}{2} du). Substituting (u) and (x , dx) into the integral, you get:

[ \begin{align*} \int (x^5)(\sqrt{4 - x^2}) , dx &= \int -\frac{1}{2} u^{\frac{1}{2}} , du \ &= -\frac{1}{2} \cdot \frac{2}{3} u^{\frac{3}{2}} + C \ &= -\frac{1}{3}(4 - x^2)^{\frac{3}{2}} + C \end{align*} ]

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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