Use substitution tUse substitution to evaluate the indefinite integral ∫3x^2e^(2x3) dx.o evaluate the indefinite integral ∫3x^2e^(2x3) dx.

Question

Use substitutioUse substitution to evaluate the indefinite integral ∫3x^2e^(2x3) dx.n to evaluate the indefinite integral ∫3x^2e^(2x3) dx.

Answer 1

To evaluate the indefinite integral ∫3x^2e^(2x^3) dx, we will use a substitution method. Let's start by making the substitution:

u = 2x^3 du/dx = 6x^2 du = 6x^2 dx

Now, we can rewrite our original function in terms of u:

∫3x^2e^(2x^3) dx = (1/2) ∫ e^u du

Now we need to find the antiderivative of e^u with respect to u:

∫ e^u du = e^u + C

Now we need to substitute back for u in terms of x:

e^(2x^3) + C

So, the antiderivative of 3x^2e^(2x^3) is:

(1/2) * e^(2x^3) + C