# Evaluate the following Integral: ∫ x(x - 4)^1/3 dx, x ∈ [4, 12]

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Evaluate the following Integral:

$\int\limits_4^{12}$x(x - 4)$\frac13$dx

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Let I =  $\int\limits_4^{12}$x(x - 4)$\frac13$dx

Put x – 4 = t3

⇒ dx = 3t2dt (Differentiating both sides)

When x = 4, t3 = 4 – 4 = 0 ⇒ t = 0

When x = 12, t3 = 12 – 4 = 8 ⇒ t = 2

So, the new limits are 0 and 2.

We can write x = t3 + 4

Substituting this in the original integral,