# How do you integrate #(x^5)*(e^(x^2/2))#?

Using Integration by Parts,

Similarly following for the problem,

Applying again integration by parts in second term, we get

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To integrate (x^5 \cdot e^{\frac{x^2}{2}}), you can use integration by parts. Let (u = x^5) and (dv = e^{\frac{x^2}{2}} dx). Then, (du = 5x^4 dx) and (v = \int e^{\frac{x^2}{2}} dx). Integrating (v), we find (v = \sqrt{2\pi} \cdot \text{erf}\left(\frac{x}{\sqrt{2}}\right)), where (\text{erf}) denotes the error function.

Applying the integration by parts formula:

[ \int u , dv = uv - \int v , du ]

We get:

[ \int x^5 \cdot e^{\frac{x^2}{2}} dx = x^5 \cdot \sqrt{2\pi} \cdot \text{erf}\left(\frac{x}{\sqrt{2}}\right) - 5 \int x^4 \cdot \sqrt{2\pi} \cdot \text{erf}\left(\frac{x}{\sqrt{2}}\right) dx ]

This expression still needs to be integrated, but now it's reduced to a simpler integral.

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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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