You first need to make the following substitution such that:
`x^3 = t => 3x^2 dx = dt => x^2 dx = (dt)/3`
You need to change variable to integrand such that:
`int x^2*e^(x^3)dx = int e^t*(dt)/3 `
Notice that you do not need to integrate by parts since you may integrate using the following formula `int e^x dx = e^x + c` .
Reasoning by analogy yields:
`(1/3) int e^t dt = (1/3)e^t + c`
Substituting back `x^3` for t yields:
`int x^2*e^(x^3)dx = (e^(x^3))/3 + c`
Hence, evaluating the given integral using substitution and not integration by parts, yields `int x^2*e^(x^3)dx = (e^(x^3))/3 + c.`