### Monkey Raptor

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## Saturday, October 11, 2014

### Math: Let's Find the Solution of ∫ 2x∛(6x - 1) dx

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 12th grade Mathematics material.

We need patience.And cool stamina
2x∛(6x - 1) dx
It's using integration by parts

Let's Start

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

Plug them into the integration-by-parts formulation variables

Continue...

Woo!

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

WINNING!

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.

Anyway
How to decide whether using integral by parts or substitution?
I usually observe the difference of the exponents.

Example

If like so:
Then use substitution.
But if like this:
Use the other method.
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.