Question #2efd1

1 Answer
Psykolord1989 .
Oct 25, 2017

U-substitution, see explanation

Explanation:

I will presume that you meant to write #int e^(x^3)x^2dx#. If you did not mean this, then the solution below will not help you.

If so, we can solve this problem via u-substitution. In u-substitution, we essentially invert the chain rule; we recognize an integrand as being an instance of #f'(g(x))*g'(x)#, and define #g(x)=u, g'(x)=du,#, allowing us to instead describe the integrand as #f'(u)du#.

In this case, if we take #x^3# to be our #u#, then #e^(x^3) = e^u, du = 3x^2#. Note that we instead have #x^2# in our integrand; this must mean that the initial function had a #1/3# before differentiation. Then we have:

#int e^(x^3)x^2dx = int 1/3e^udu#

By the definition of the derivative and integral of #e^x#, this becomes:

#= 1/3 e^u +c#

Putting the equation back in terms of x...

#= 1/3 e^(x^3)+c#

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