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

Answer 1

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.

Let #u=x^3#
#int_0^1x^2e^udx#
#du=3x^2dx#
#(du)/3=x^2dx#
#1/3*du=x^2dx#
#inte^ux^2dx#, notice that #x^2dx# can be replaced by #1/3*du#
#inte^u1/3*du#
#1/3inte^u*du#, Constants can be moved outside of the integral
Now lets evaluate the boundaries. Look back to the original #u# substitution: #u=x^3#

Substitute in the current high and low boundaries.

#u=(1)^3=1 -># upper boundary #u=(0)^3=0 -># lower boundary

In this problem the boundaries did not change

#1/3int_0^1e^u*du#
#=1/3[e^u]_0^1=1/3[e^1-e^0]=1/3[e^1-1]=(e^1-1)/3=(e-1)/3#
#=0.5728#
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Answer 2

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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Answer from HIX Tutor

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.

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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