I'm having trouble understanding what i'm doing here to get the answer.

For instance

Find indefinite integral for x^3cos((x^4)+2)dx

ok first of all, what is the dx and why is it there at the end?

To do this problem from the instructions, I let

u = (x^4 ) + 2

du = 4x^3 dx

Is du just the derivative of u? Also why is dx there again and what does it represent?

from here I would

x^3 dx = (1/4)du

Now that I have those, according to the substitution rule, it should be

(cos u)((1/4)du) ok...now this is the part I don't understand even more is

for Cos u, you find the antiderivative and get Sin u where u will be replaced with (x^4) + 2 giving you Sin((x^4) +2). That part is easily understandable. The part I don't understand is why is ((1/4)du) just = 1/4? where did the du go? U was replaced easily with ((x^4) +2) so why can't du be replaced with 4x^3 but instead it's replaced with 1/4? To my understanding, the Substitution Rule formula is f(u) du and this means f(u) times the derivative of u. so the equation should look like

cos((x^4) + 2) multiplied by 4x^3. Apparently somewhere along in there i'm missing something and I don't get it. I'm interpreting the formula exactly the way it's written and can't get an understanding of what steps i'm skipping or missing.

Any help is appreciated.