Math: ∫ 2x∛(6x - 1) dx

I forgot to do it properly. So I put here as my own reminder. Here goes.

$∫ 2x∛(6x - 1) dx$

Integration by Parts

$∫ u.dv = u.v - ∫ v du$

Start

$u = 2x$

$dv = ∛(6x - 1) dx$

We need the $du$ and $v$:

${du}/{dx} = 2$

$du = 2 dx$

$v = ∫ ∛(6x - 1) dx$

$= ∫ (6x - 1)^{1/3} dx$

$= {1/6} × 1/{1/3 + 1} × (6x - 1)^{1/3 + 1}$

$= 1/6 × 1/{4/3} × (6x - 1)^{4/3}$

$= 1/6 × 3/4 × (6x - 1)^{4/3}$

$= 1/8 × (6x - 1)^{4/3}$

We have these variables: u — du — dv — v.

Let's use images then, as you can see below:

---

Plug

---

Continue

👍

---

Another Form of Solution

Expand it once again.

👍

---

Other Technique

We can also use tabular integration by parts method.

It's faster.

---

How to decide whether using integral by parts or substitution?

I usually observe the difference of the exponents.

If like so:

Then use substitution.

But if like this:

Use integration by parts.