Monkey Raptor

I forgot to do it properly. And my brain crashed for a moment, a while ago. Therefore, I re-re-re-tinkered it on my scribble pad.
What do you know? It's not THAT complicated.

Anyway, this is a 12^th grade Mathematics material.

We need patience._{And cool stamina}

∫ 2x∛(6x - 1) dx

It's using integration by parts
∫ u dv = u.v - ∫ v du

Let's Start
We take the variables u and dv like so: u = 2x dv = ∛(6x - 1) dx *** Then we need the du and v: du/dx (derivative of u with respect to x) = 2 (switch the dx to right side to get du) ► du = 2 dx v (integral of dv with respect to x) = ∫ ( ∛(6x - 1) ) dx = ∫ (6x - 1)^(1/3) dx = (1/6) * 1/(1/3 + 1) * (6x - 1)^(1/3 + 1) = 1/6 * 1/(4/3) * (6x - 1)^(4/3) = 1/6 * 3/4 * (6x - 1)^(4/3) = 1/8 * (6x - 1)^(4/3)

So then, we have bunch of variables, u — du — dv — v.
Let's use images then, as you can see below:

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Plug them into the integration-by-parts formulation variables

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Continue...

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Probably we need another form of solution
Let's expand it once moar

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Other method
We can also use tabular integration by parts method.
It's faster and much more organized, I suppose.
It's a great method for longer form which can be solved using integration by parts.
You can try using it by yourself, and see if the result will be the same as those two above.

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Anyway

How to decide whether using integral by parts or substitution?

I usually observe the difference of the exponents.

Example

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If like so:

Then use substitution.

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But if like this:

Use the other method.

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But then again, it depends on the problem. By doing problem sets repetitions, you'll have the "basic instinct".
Just like sport. But mostly with the conscious brain.

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Link
Tabular integration by parts at Wikipedia