Integration by substitution

Question

What u expression can be used to complete the following integrals by substitution? (i.e. u = ?)

integral of x^2*(e^x)^3

Answer 1

Write the expression as x²e^{x^3} if that is what you mean. As you wrote it, x²(e^x)³ = x²e^3x which is not solved by substitution, but by integration by parts twice.

Answer 2

Let u = x³. So, du = 3x²dx. Therefore, x²dx = (1/3)du

∫x²(e^x)³dx = (1/3)∫e^udu = (1/3)e^u + C = (1/3)(e^x)³ + C