Find the integral of (3x^2)(e^(x^3)) dx.

To find the integral of ( (3x^2)(e^{x^3}) , dx ), we can use integration by substitution.

Let us substitute ( u = x^3 ). Then, the differential ( du = 3x^2 , dx ). This means ( dx = \frac{du}{3x^2} ). We can express ( 3x^2 , dx ) simply as ( du ).

Rewriting the integral in terms of ( u ), we have:

[

\int (3x^2)(e^{x^3}) , dx = \int e^{u} , du

]

The integral of ( e^{u} ) is simply ( e^{u} + C ), where ( C ) is the constant of integration. Now substituting back ( u = x^3 ):

[

e^{u} + C = e^{x^3} + C

]

This leads us directly to the correct answer.

Therefore, the correct choice is based on the fact that when integrating ( 3x^2 e^{x^3} ), we seamlessly transition through the substitution and arrive