# How do you find the integral #int_0^1x^2*e^(x^3)dx# ?

We have to use a substitution technique to solve this problem. The strategy is to find an expression that when then differentiated can be substituted back into the original integral.

Substitute in the current high and low boundaries.

In this problem the boundaries did not change

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To find the integral of ( \int_{0}^{1} x^2 \cdot e^{x^3} , dx ), you can use the technique of substitution. Let ( u = x^3 ), then ( du = 3x^2 , dx ). Rearrange this equation to solve for ( dx ), yielding ( dx = \frac{1}{3x^2} , du ). Now, substitute ( u ) and ( dx ) in terms of ( x ) into the integral:

[ \int_{0}^{1} x^2 \cdot e^{x^3} , dx = \int_{0}^{1} \frac{1}{3} e^u , du ]

Now, integrate ( e^u ) with respect to ( u ):

[ = \frac{1}{3} \int_{0}^{1} e^u , du = \frac{1}{3} \left[ e^u \right]_{0}^{1} ]

[ = \frac{1}{3} \left( e^1 - e^0 \right) = \frac{1}{3} (e - 1) ]

So, ( \int_{0}^{1} x^2 \cdot e^{x^3} , dx = \frac{1}{3} (e - 1) ).

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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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