How do you integrate #x^4 e^(x^5) dx#?
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To integrate (x^4 e^{x^5} , dx), you can use the substitution method:

Let ( u = x^5 ), then ( du = 5x^4 , dx ), so ( dx = \frac{du}{5x^4} ).

Substitute ( u = x^5 ) and ( dx = \frac{du}{5x^4} ) into the integral.

The integral becomes ( \frac{1}{5} \int u e^u , du ).

Now integrate ( \int u e^u , du ) using integration by parts.

Let ( dv = e^u , du ), then ( v = e^u ) and ( du = du ).

Apply the integration by parts formula: ( \int u , dv = uv  \int v , du ).

Substitute ( u ) and ( dv ) and integrate accordingly.

Finally, substitute back ( u = x^5 ) to get the result.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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