Evaluate the following integral: `int_4^(12)x(x-4)^(1//3)dx`

Explanation

Given,`int_4^(12)x(x-4)^(1//3)dx`

Put `x – 4 = t^3`

⇒ `dx = 3t^2dt` (Differentiating both sides)
When x = 4, `t^3 = 0 ⇒ t = 0`

When x = 12, `t^3 = 12 – 4 = 8 ⇒ t = 2`

So, the new limits are 0 and 2.

We can write `x = t^3 + 4`
Substituting this in the original integral

I=`int_0^(2)( t^3 + 4)t 3t^2dt`

I=`3int_0^(2)( t^6 + 4t^3)dt`

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