Question

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`