# How do you use substitution to integrate #x^2sqrt(x^(4)+5)#?

If I rewrite this as:

I think you can do it from there (I'm in a hurry. If someone wants to finish this, go ahead).

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To integrate (x^2\sqrt{x^4 + 5}) using substitution, let (u = x^4 + 5). Then, (du = 4x^3 dx). Rearranging gives (x^3 dx = \frac{1}{4} du). Now, substitute:

[u = x^4 + 5] [du = 4x^3 dx] [x^3 dx = \frac{1}{4} du]

So, the integral becomes:

[\int x^2\sqrt{x^4 + 5} , dx = \int \frac{1}{4} \sqrt{u} , du]

Now, you can integrate (\frac{1}{4} \sqrt{u}) with respect to (u). Once you find the antiderivative, don't forget to substitute back (x) for (u).

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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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