# How do you integrate #int x5^x dx# using integration by parts?

Via integration by parts:

In the case of

We set

We plug these values in to see that

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To integrate ( \int x^5 \cdot e^x , dx ) using integration by parts, you can follow these steps:

- Choose ( u ) and ( dv ).
- Compute ( du ) and ( v ).
- Apply the integration by parts formula:

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

Here's how to apply these steps to ( \int x^5 \cdot e^x , dx ):

- Choose ( u = x^5 ) and ( dv = e^x , dx ).
- Compute ( du = 5x^4 , dx ) and ( v = e^x ).
- Apply the integration by parts formula:

[ \int x^5 \cdot e^x , dx = x^5 \cdot e^x - \int 5x^4 \cdot e^x , dx ]

Now, the integral ( \int 5x^4 \cdot e^x , dx ) is of a similar form as the original integral, so you can apply integration by parts again. Repeat the process until you reach an integral that can be easily evaluated.

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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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