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

1 Answer
Sep 22, 2014

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#